introduction to the quadratic formula assignment

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Study Guides > Boundless Algebra

Introduction to quadratic functions, what is a quadratic function, learning objectives, key takeaways.

• A quadratic function is of the form [latex]f(x)=ax^2+bx+c[/latex], where a is a nonzero constant, b and c are constants of any value, and x is the independent variable.
• The solutions to a quadratic equation are known as its zeros, or roots.
• dependent variable : Affected by a change in input, i.e. it changes depending on the value of the input.
• independent variable : The input of a function that can be freely varied.
• vertex : The minimum or maximum point of a quadratic function.
• quadratic function : A function of degree two.

Differences Between Quadratics and Linear Functions

• Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope). All quadratic functions both increase and decrease.
• With a linear function, each input has an individual, unique output (assuming the output is not a constant).  With a quadratic function, pairs of unique independent variables will produce the same dependent variable, with only one exception (the vertex ) for a given quadratic function.
• The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

Forms of Quadratic Functions

The quadratic formula.

• The quadratic formula is: [latex]x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}[/latex], where [latex]a[/latex] and [latex]b[/latex] are the coefficients of the [latex]x^2[/latex] and [latex]x[/latex] terms, respectively, in a quadratic equation, and [latex]c[/latex] is the value of the equation's constant.
• To use the quadratic formula, [latex]ax^2 + bx+c [/latex] must equal zero and [latex]a[/latex] must not be zero.
• zero : Also known as a root, an [latex]x[/latex] value at which the function of [latex]x[/latex] is equal to 0.

Criteria For Use

• The quadratic equation must equal zero;  [latex]ax^2+bx+c=0[/latex]
• [latex]a[/latex] must not equal zero

Solving Quadratic Equations with the Quadratic Formula

The discriminant.

• [latex]\Delta =b^2-4ac[/latex] is the formula for a quadratic function 's discriminant.
• If Δ is greater than zero, the polynomial has two real, distinct roots.
• If Δ is equal to zero, the polynomial has only one real root.
• If Δ is less than zero, the polynomial has no real roots, only two distinct complex roots.
• A zero is the x value whereat the function crosses the x-axis. That is, it is the x-coordinate at which the function's value equals zero.
• quadratic : Of degree two; can apply to polynomials.
• zero : Also known as a root; an x value at which the function of x is equal to zero.
• discriminant : An expression that gives information about the roots of a polynomial.

The Discriminant and the Quadratic Formula

Positive discriminant, zero discriminant, negative discriminant.

Other Equations in Quadratic Form

• A biquadratic equation (quartic equation with no terms of odd- degree ) has the form [latex]0=ax^4+bx^2+c[/latex]. It can be expressed as: [latex]0=ap^2+bp+c[/latex] (where [latex]p=x^2[/latex]).
• The values of [latex]p[/latex] can be found by graphing, factoring, completing the square, or using the quadratic formula. Their square roots (positive and negative) are the values of [latex]x[/latex] that satisfy the original equation.
• Higher-order equations can be solved by a similar process that involves reducing their exponents. The requirement is that there are two terms of [latex]x[/latex] such that the ratio of the highest exponent of [latex]x[/latex] to the lower is [latex]2:1[/latex].
• zero : Also known as a root; an [latex]x[/latex] value at which the function of [latex]x[/latex] is equal to zero.
• biquadratic : When a polynomial involves only the second and fourth powers of a variable.
• quartic function : Any polynomial function whose greatest exponent is of power four.

We can substitute the arbitrary variable [latex]p[/latex] in place of [latex]x^2[/latex]:

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