problem solving using quadratic equations worksheet

Solving Quadratic Equations: Worksheets with Answers

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Quadratic Formula Exercises

Quadratic formula practice problems with answers.

Below are ten (10) practice problems regarding the quadratic formula. The more you use the formula to solve quadratic equations, the more you become expert at it!

Use the illustration below as a guide. Notice that in order to apply the quadratic formula, we must transform the quadratic equation into the standard form, that is, [latex]a{x^2} + bx + c = 0[/latex] where [latex]a \ne 0[/latex].

The problems below have varying levels of difficulty. I encourage you to try them all. Believe me, they are actually easy! Good luck.

Problem 1: Solve the quadratic equation using the quadratic formula.

[latex]{x^2}\, – \,8x + 12 = 0[/latex]

Therefore, the answers are [latex]{x_1} = 6[/latex] and [latex]{x_2} = 2[/latex].

Problem 2: Solve the quadratic equation using the quadratic formula.

[latex]2{x^2}\, -\, x = 1[/latex]

Rewrite the quadratic equation in the standard form.

[latex]2{x^2} – x – 1 = 0[/latex]

Therefore, the answers are [latex]{x_1} = 1[/latex] and [latex]{x_2} = \large{{ – 1} \over 2}[/latex].

Problem 3: Solve the quadratic equation using the quadratic formula.

[latex]4{x^2} + 9 = – 12x[/latex]

[latex]4{x^2} + 12x + 9 = 0[/latex]

Therefore, the solution is [latex]x = \large{{ – 3} \over 2}[/latex].

Problem 4: Solve the quadratic equation using the quadratic formula.

[latex]5{x^2} = 7x + 6[/latex]

Convert the quadratic equation into the standard form.

[latex]5{x^2} – 7x – 6 = 0[/latex]

Therefore, the answers are [latex]{x_1} = 2[/latex] and [latex]{x_2} = \large{{ – 3} \over 5}[/latex].

Problem 5: Solve the quadratic equation using the quadratic formula.

[latex]{x^2} -\,{ \large{1 \over 2}}x\, – \,{\large{3 \over {16}}} = 0[/latex]

Multiply the entire equation by the LCM of the denominators which is [latex]16[/latex]. This will get rid of the denominators thereby giving us integer values for [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex].

[latex]16{x^2} – 8x – 3 = 0[/latex]

Therefore, the answers are [latex]x_1=\large{3 \over 4}[/latex] and [latex]x_2=\large{{ – 1} \over 4}[/latex].

Problem 6: Solve the quadratic equation using the quadratic formula.

[latex]{x^2} + 3x + 9 = 5x – 8[/latex]

Convert into standard form as [latex]{x^2} – 2x + 17 = 0[/latex].

Therefore, the answers are [latex]x_1=1 + 4i[/latex] and [latex]x_2=1 – 4i[/latex].

Problem 7: Solve the quadratic equation using the quadratic formula.

[latex]{\left( {x – 2} \right)^2} = 4x[/latex]

Rewrite in standard form as [latex]{x^2} – 8x + 4 = 0[/latex].

Hence, the answers are [latex]{x_1} = 4 + 2\sqrt 3 [/latex] and [latex]{x_2} = 4 – 2\sqrt 3 [/latex].

Problem 8: Solve the quadratic equation using the quadratic formula.

[latex]{\Large{{{x^2}} \over 4} – {x \over 2} }= 1[/latex]

To convert the quadratic equation into the standard form, simply multiply the entire equation by [latex]4[/latex] then subtract both sides by [latex]4[/latex].

[latex]{x^2} – 2x – 4 = 0[/latex]

Thus, the answers are [latex]{x_1} = 1 + \sqrt 5 [/latex] and [latex]{x_2} = 1 – \sqrt 5 [/latex].

Problem 9: Solve the quadratic equation using the quadratic formula.

[latex]{\left( {2x – 1} \right)^2} = \Large{x \over 3}[/latex]

If we carefully transform the given quadratic equation into the standard form, we get [latex]12{x^2} – 13x + 3 = 0[/latex].

Therefore, the answers are [latex]x_1={\Large{3 \over 4}}[/latex] and [latex]x_2={\Large{1 \over 3}}[/latex].

Problem 10: Solve the quadratic equation using the quadratic formula.

[latex]\left( {2x – 1} \right)\left( {x + 4} \right) = – {x^2} + 3x[/latex]

If we simplify the quadratic equation to convert it to the standard form, we should arrive at [latex]3{x^2} + 4x – 4 = 0[/latex].

Hence, the answers are [latex]x_1={\Large{2 \over 3}}[/latex] and [latex]x_2=-2[/latex].

You may also be interested in these related math lessons or tutorials:

The Quadratic Formula

Solving Quadratic Equations using the Quadratic Formula

Quadratic Formula Worksheets

Related Topics: More Math Worksheets More Grade 7 Math Lessons Grade 7 Math Worksheets

There are two sets of solving quadratic equation worksheets:

• Solve Quadratic Equation (use factoring)
• Solve Quadratic Equation (use quadratic formula)

Examples, solutions, videos, and worksheets to help Grade 7 and Grade 8 students learn how to solve quadratic equations using the quadratic formula.

How to solve quadratic equations using the quadratic formula?

There are three sets of solving equations using the quadratic formula.

• Quadratic Formula (rational solutions).
• Quadratic Formula (irrational solutions).
• Quadratic Formula (complex solutions).

These are the steps to solve a quadratic equation using the quadratic formula:

• Write Down the Equation: Start with the quadratic equation in the standard form ax 2 + bx + c = 0, where a, b, and c are constants.
• Identify Coefficients: Identify the values of a, b, and c from the quadratic equation.
• Apply the Quadratic Formula: Use the quadratic formula to calculate the solutions of the equation:
• There are two possible solutions, one with the positive square root and one with the negative square root (
• Calculate Solutions: Substitute the values of a, b, and c into the formula and calculate the solutions for x.

The quadratic formula provides an accurate method to solve any quadratic equation, whether it has real or complex solutions.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Quadratic Formula Worksheets .

More Quadratic Formula Worksheets

Printable (Answers on the second page.) Quadratic Formula Worksheet #1 (rational solutions) Quadratic Formula Worksheet #2 (irrational solutions) Quadratic Formula Worksheet #3 (complex solutions)

Online or Generated Factor Binomials by Difference of Squares Factor Perfect Square Trinomials Factor Trinomials or Quadratic Equations Factor Different Types of Trinomials 1 Factor Different Types of Trinomials 2 Solve Trinomials using Quadratic Formula Find Discriminants of Quadratic Polynomials

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Quadratic Formula Solve Equations using Quadratic Formula

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9.4: Solve Quadratic Equations Using the Quadratic Formula

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Learning Objectives

By the end of this section, you will be able to:

• Solve quadratic equations using the Quadratic Formula
• Use the discriminant to predict the number and type of solutions of a quadratic equation
• Identify the most appropriate method to use to solve a quadratic equation

Before you get started, take this readiness quiz.

• Evaluate \(b^{2}-4 a b\) when \(a=3\) and \(b=−2\).
• Simplify \(\sqrt{108}\).
• Simplify \(\sqrt{50}\).

Solve Quadratic Equations Using the Quadratic Formula

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.

We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for \(x\) .

We start with the standard form of a quadratic equation and solve it for \(x\) by completing the square.

The final equation is called the "Quadratic Formula."

Definition \(\PageIndex{1}\): Quadratic Formula

The solutions to a quadratic equation of the form \(a x^{2}+b x+c=0\), where \(a≠0\) are given by the formula:

\[x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \label{quad}\]

To use the Quadratic Formula , we substitute the values of \(a,b\), and \(c\) from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.

Notice the Quadratic Formula (Equation \ref{quad}) is an equation. Make sure you use both sides of the equation.

Example \(\PageIndex{1}\) How to Solve a Quadratic Equation Using the Quadratic Formula

Solve by using the Quadratic Formula: \(2 x^{2}+9 x-5=0\).

Exercise \(\PageIndex{1}\)

Solve by using the Quadratic Formula: \(3 y^{2}-5 y+2=0\).

\(y=1, y=\dfrac{2}{3}\)

Exercise \(\PageIndex{2}\)

Solve by using the Quadratic Formula: \(4 z^{2}+2 z-6=0\).

\(z=1, z=-\dfrac{3}{2}\)

HowTo: Solve a Quadratic Equation Using the Quadratic Formula

• Write the quadratic equation in standard form, \(a x^{2}+b x+c=0\). Identify the values of \(a,b\), and \(c\).
• Write the Quadratic Formula. Then substitute in the values of \(a,b\), and \(c\).
• Check the solutions.

If you say the formula as you write it in each problem, you’ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with “\(x=\)”.

Example \(\PageIndex{2}\)

Solve by using the Quadratic Formula: \(x^{2}-6 x=-5\).

Exercise \(\PageIndex{3}\)

Solve by using the Quadratic Formula: \(a^{2}-2 a=15\).

\(a=-3, a=5\)

Exercise \(\PageIndex{4}\)

Solve by using the Quadratic Formula: \(b^{2}+24=-10 b\).

\(b=-6, b=-4\)

When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula . If we get a radical as a solution, the final answer must have the radical in its simplified form.

Example \(\PageIndex{3}\)

Solve by using the Quadratic Formula: \(2 x^{2}+10 x+11=0\).

Exercise \(\PageIndex{5}\)

Solve by using the Quadratic Formula: \(3 m^{2}+12 m+7=0\).

\(m=\dfrac{-6+\sqrt{15}}{3}, m=\dfrac{-6-\sqrt{15}}{3}\)

Exercise \(\PageIndex{6}\)

Solve by using the Quadratic Formula: \(5 n^{2}+4 n-4=0\).

\(n=\dfrac{-2+2 \sqrt{6}}{5}, n=\dfrac{-2-2 \sqrt{6}}{5}\)

When we substitute \(a, b\), and \(c\) into the Quadratic Formula and the radicand is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.

Example \(\PageIndex{4}\)

Solve by using the Quadratic Formula: \(3 p^{2}+2 p+9=0\).

Exercise \(\PageIndex{7}\)

Solve by using the Quadratic Formula: \(4 a^{2}-2 a+8=0\).

\(a=\dfrac{1}{4}+\dfrac{\sqrt{31}}{4} i, \quad a=\dfrac{1}{4}-\dfrac{\sqrt{31}}{4} i\)

Exercise \(\PageIndex{8}\)

Solve by using the Quadratic Formula: \(5 b^{2}+2 b+4=0\).

\(b=-\dfrac{1}{5}+\dfrac{\sqrt{19}}{5} i, \quad b=-\dfrac{1}{5}-\dfrac{\sqrt{19}}{5} i\)

Remember, to use the Quadratic Formula, the equation must be written in standard form, \(a x^{2}+b x+c=0\). Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.

Example \(\PageIndex{5}\)

Solve by using the Quadratic Formula: \(x(x+6)+4=0\).

Our first step is to get the equation in standard form.

Exercise \(\PageIndex{9}\)

Solve by using the Quadratic Formula: \(x(x+2)−5=0\).

\(x=-1+\sqrt{6}, x=-1-\sqrt{6}\)

Exercise \(\PageIndex{10}\)

Solve by using the Quadratic Formula: \(3y(y−2)−3=0\).

\(y=1+\sqrt{2}, y=1-\sqrt{2}\)

When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions— to solve. We can use the same strategy with quadratic equations.

Example \(\PageIndex{6}\)

Solve by using the Quadratic Formula: \(\dfrac{1}{2} u^{2}+\dfrac{2}{3} u=\dfrac{1}{3}\).

Our first step is to clear the fractions.

Exercise \(\PageIndex{11}\)

Solve by using the Quadratic Formula: \(\dfrac{1}{4} c^{2}-\dfrac{1}{3} c=\dfrac{1}{12}\).

\(c=\dfrac{2+\sqrt{7}}{3}, \quad c=\dfrac{2-\sqrt{7}}{3}\)

Exercise \(\PageIndex{12}\)

Solve by using the Quadratic Formula: \(\dfrac{1}{9} d^{2}-\dfrac{1}{2} d=-\dfrac{1}{3}\).

\(d=\dfrac{9+\sqrt{33}}{4}, d=\dfrac{9-\sqrt{33}}{4}\)

Think about the equation \((x-3)^{2}=0\). We know from the Zero Product Property that this equation has only one solution, \(x=3\).

We will see in the next example how using the Quadratic Formula to solve an equation whose standard form is a perfect square trinomial equal to \(0\) gives just one solution. Notice that once the radicand is simplified it becomes \(0\), which leads to only one solution.

Example \(\PageIndex{7}\)

Solve by using the Quadratic Formula: \(4 x^{2}-20 x=-25\).

Did you recognize that \(4 x^{2}-20 x+25\) is a perfect square trinomial. It is equivalent to \((2 x-5)^{2}\)? If you solve \(4 x^{2}-20 x+25=0\) by factoring and then using the Square Root Property, do you get the same result?

Exercise \(\PageIndex{13}\)

Solve by using the Quadratic Formula: \(r^{2}+10 r+25=0\).

Exercise \(\PageIndex{14}\)

Solve by using the Quadratic Formula: \(25 t^{2}-40 t=-16\).

\(t=\dfrac{4}{5}\)

Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?

Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant .

Definition \(\PageIndex{2}\)

Discriminant

Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.

Using the Discriminant \(b^{2}-4ac\), to Determine the Number and Type of Solutions of a Quadratic Equation

For a quadratic equation of the form \(ax^{2}+bx+c=0\), \(a \neq 0\),

• If \(b^{2}-4 a c>0\), the equation has \(2\) real solutions.
• if \(b^{2}-4 a c=0\), the equation has \(1\) real solution.
• if \(b^{2}-4 a c<0\), the equation has \(2\) complex solutions.

Example \(\PageIndex{8}\)

Determine the number of solutions to each quadratic equation.

\(3 x^{2}+7 x-9=0\)

\(5 n^{2}+n+4=0\)

\(9 y^{2}-6 y+1=0\)

To determine the number of solutions of each quadratic equation, we will look at its discriminant.

The equation is in standard form, identify \(a, b\), and \(c\).

\(a=3, \quad b=7, \quad c=-9\)

Write the discriminant.

\(b^{2}-4 a c\)

Substitute in the values of \(a, b\), and \(c\).

\((7)^{2}-4 \cdot 3 \cdot(-9)\)

\(49+108\) \(157\)

Since the discriminant is positive, there are \(2\) real solutions to the equation.

\(a=5, \quad b=1, \quad c=4\)

\((1)^{2}-4 \cdot 5 \cdot 4\)

\(1-80\) \(-79\)

Since the discriminant is negative, there are \(2\) complex solutions to the equation.

\(a=9, \quad b=-6, \quad c=1\)

\((-6)^{2}-4 \cdot 9 \cdot 1\)

\(36-36\) \(0\)

Since the discriminant is \(0\), there is \(1\) real solution to the equation.

Exercise \(\PageIndex{15}\)

Determine the number and type of solutions to each quadratic equation.

• \(8 m^{2}-3 m+6=0\)
• \(5 z^{2}+6 z-2=0\)
• \(9 w^{2}+24 w+16=0\)
• \(2\) complex solutions
• \(2\) real solutions
• \(1\) real solution

Exercise \(\PageIndex{16}\)

• \(b^{2}+7 b-13=0\)
• \(5 a^{2}-6 a+10=0\)
• \(4 r^{2}-20 r+25=0\)

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We summarize the four methods that we have used to solve quadratic equations below.

Methods for Solving Quadratic Equations

• Square Root Property
• Completing the Square
• Quadratic Formula

Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is \(ax^{2}=k\) or \(a(x−h)^{2}=k\) we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.

What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.

Identify the Most Appropriate Method to Solve a Quadratic Equation

• Try Factoring first. If the quadratic factors easily, this method is very quick.
• Try the Square Root Property next. If the equation fits the form \(ax^{2}=k\) or \(a(x−h)^{2}=k\), it can easily be solved by using the Square Root Property.
• Use the Quadratic Formula . Any other quadratic equation is best solved by using the Quadratic Formula.

The next example uses this strategy to decide how to solve each quadratic equation.

Example \(\PageIndex{9}\)

Identify the most appropriate method to use to solve each quadratic equation.

• \(5 z^{2}=17\)

\(4 x^{2}-12 x+9=0\)

\(8 u^{2}+6 u=11\)

\(5z^{2}=17\)

Since the equation is in the \(ax^{2}=k\), the most appropriate method is to use the Square Root Property.

We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.

Put the equation in standard form.

\(8 u^{2}+6 u-11=0\)

While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.

Exercise \(\PageIndex{17}\)

• \(x^{2}+6 x+8=0\)
• \((n-3)^{2}=16\)
• \(5 p^{2}-6 p=9\)

Exercise \(\PageIndex{18}\)

• \(8 a^{2}+3 a-9=0\)
• \(4 b^{2}+4 b+1=0\)
• \(5 c^{2}=125\)
• Factoring or Square Root Property

Access these online resources for additional instruction and practice with using the Quadratic Formula.

• Using the Quadratic Formula
• Solve a Quadratic Equation Using the Quadratic Formula with Complex Solutions
• Discriminant in Quadratic Formula

Key Concepts

\(x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

• Write the quadratic equation in standard form, \(a x^{2}+b x+c=0\). Identify the values of \(a, b, c\).
• Write the Quadratic Formula. Then substitute in the values of \(a, b, c\).
• If \(b^{2}-4 a c=0\), the equation has \(1\) real solution.
• If \(b^{2}-4 a c<0\), the equation has \(2\) complex solutions.
• Try the Square Root Property next. If the equation fits the form \(a x^{2}=k\) or \(a(x-h)^{2}=k\), it can easily be solved by using the Square Root Property.
• Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Quadratic Formula Worksheets (pdfs)

Free worksheets with answer keys.

Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Quadratic Formula Worksheet. Real Solutions
• Quadratic Formula Worksheet. Complex Solutions
• Quadratic Formula Worksheet. Both Real and Complex Solutions

Quadratic Equations Factoring Worksheets

Quadratic equations, factoring: monic, factoring: non-monic.

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Quadratic Equation Word Problems

Take the young mathematician in you on a jaunt to this printable compilation of quadratic word problems and discover the role played by quadratic equations inspired from a variety of real-life scenarios! From finding the area of your small playroom to calculating the speed of a massive cruise, quadratic equations matter a lot in life. Try this simple question: Alan is 2 years older than Clara. If the product of both Allan’s and Clara’s ages is 168, how old is Clara? As soon as you read this, this equation will ring a bell: x(x + 2) = 168. If you rearrange and rewrite this, you'll have x 2 + 2x - 168 = 0. Solve this equation to obtain their ages. As far as this problem is concerned, Alan is 14 years and Clara is 12 years. Now, print our worksheet pdfs, exclusively designed for high school students and get to solve 15 similar word problems. Read each word problem, formulate a quadratic equation, and solve for the unknown. You can use any of these methods: factoring, square roots, completing squares, or quadratic formula to arrive at your answers.

• Download the set

Related Worksheets

» Quadratic Equation | Factoring

» Quadratic Equation | Completing Squares

» Quadratic Equation | Square Root

» Quadratic Formula

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Quadratic Word Problems Worksheets

In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms.

In the quadratic equations word problems, the equations wouldn't be given directly. In fact, you have to deduct the equation from the given facts within the equations. It can also include profit maximization or loss minimization questions in which you have to find either minimum or maximum value of the equation.

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Solving Quadratic Equations - Worded problems worksheet with grades

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

23 December 2020

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Worded quadratic equation problems requiring students to decompose a problem, form a quadratic equation, factorise it and solve it. They then have to choose a solution that matches the context of the question. This is the FREE version of the popular ( 8k+ downloads and TES Picks ) resource with a 2 x Grade 5 questions and solutions. The full resource can be found here with 7 x Grade 5 - Grade 7 questions. It’s ideal for a 45-60 minute lesson and students can select what they want to do.

WANT TO PREPARE STUDENTS TO FACTORISE QUADRATICS? Why not take a look at this graded worksheet with examples on how to factorise quadratics using the box method.

WANT TO HELP STUDENTS GRASP COMPLETING THE SQUARE? Why not take a look at this visual method resource which provides step-by-step guidance and questions on how to complete the square .

WANT TO BUY A QUADRATICS BUNDLE AND SAVE MONEY AND TIME? Take a look at this factorising quadratics, solving quadratic equations and completing the square bundle .

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