Quadratic Equations / Functions

Content

• Introduction
• Solving Quadratic Equations by Factoring
• Greatest Common Factor
• Square Root Property
• Foil
• Difference of Two Squares
• Factor by Grouping
• Solving Quadratic Equations by Completing the Square
• Solving Quadratic Equations with the Quadratic Formula
• Using and Understanding the Discriminant

Introduction

Quadratic functions are easy to recognize. The polynomial expression known as a quadratic contains a variable that is squared making it a 2nd degree equation, and the graph is U-shaped. Quadratic expressions that are equal to zero are called quadratic equations.

The standard form of a quadratic equation is: $ax^{2} + bx + c = 0$

The graph of a quadratic equation has a recognizable shape – a parabola. The parabola may open up or down, and the direction of the opening is determined by the sign of the leading coeeficient.

Solving Quadratic Equations by Factoring

To find the solutions to quadratic equations, also known as the zeros or roots, set the quadratic expression equal to zero then factor. The values for x identify where the graph touches the x-axis. There are several methods you can use to factor quadratic equations.

Greatest Common Factor

Identify the greatest common factor (GCF) of all the terms in the quadratic expression and use the reverse of the Distributive Property to factor.

This equation has a GCF equal to 2x.

$\begin{align} 2x^{2} + 2x& = 0\\ 2x(x + 2) &= 0 \end{align}$

$x = -2, 0$

The graph touches the x-axis at -1 and 0.

Square Root Property

Use the property of square roots to find the zeros of quadratic equations such as this one.

$\begin{align} x^{2} - 36 &= 0\\ x^{2} - 36 + 36 &= 0 +36\\ x^{2} &= 36\\ x&=\sqrt{36}\\ \sqrt{36} &=\pm 6 \end{align}$

The solution to the quadratic equation is -6 and 6.

Foil

The foil method is used to simplify the product of two binomials, so the reverse of the foil method can be used to factor quadratic equations with trinomial expressions having a equal to 1.

To reverse the foil method, find factors of the c that sum to the b.

$ax^{2} + bx + c = 0$
$x^{2} + 7x + 6 = 0$

For the product of 6, the factors 1 and 6 sum to 7. Inside two sets of parentheses, add the constants of 6 and 1 to x respectively then set each binomial equal to zero and solve to determine the roots of the equations.

$(x + 6)(x+ 1) = 0$
$x = -6, -1$

To check your work, FOIL.

$\begin{align} (x + 6)(x+ 1)&=0\\ x^{2} + 6x + 1x +6&=0\\ x^{2} + 7x +6&=0 \end{align}$

The roots of the equation are -6 and -1.

Difference of Two Squares

A quadratic equation that is the difference of two squares is also known as a DOTS equation. If you can recognize which quadratic equations are DOTS (difference of two squares), you can save yourself time when factoring quadratic equations.

To identify DOTS, look for a specific pattern in the quadratic equation. Notice there are only two terms and both are perfect squares. The solution is the square root of the constant.

$ax^{2} + bx + c = 0$

$\begin{align} x^{2} - 49 & =0\\ (x +7) (x -7)&=0 \end{align}$

The solution to this DOTS equations is -7 and 7.

Factor by Grouping

When the quadratic equation has a trinomial expression with $a\neq 1$, you can factor by grouping. There are several steps to this method.

$ax^{2} + bx + c = 0$
$2x^{2} + -6x -8 =0$

Factor by Grouping

1. Find the product of a and c.
2. Identify two factors that sum to b.
3. Write new values for bx.
4. Group the factors using parentheses
5. Factor out the GCF of each group
6. Set up the binomial factors.

For this equation $a\times c = -16$. -2 and 8 sum to -6. Take a look at the next steps to solve this quadratic equation.

$\begin{align} 2x^{2} -6x -8 & =0\\ 2x^{2} +2x -8x -8 & =0\\ (2x^{2} +2x)+ (-8x - 8)&=0\\ 2x(x +1) -8(x +1)&=0\\ (2x -8)(x +1)&=0 \end{align}$

The solution to the equation is -1 and 4.

Solving Quadratic Equations by Completing the Square

When you are unable to determine factors, you can use the complete the square method to solve quadratic equations. To determine the roots using this method, there are several steps.

$ax^{2} + bx + c = 0$

1. Use the opposite operation to move the constant to the other side of the standard form.
2. Take half of b, square it and add to both sides of the equation.
3. Factor the perfect square on left side of the equation.
4. Apply the square root property to solve.

$\begin{align} x^{2} + 2x -7 &= 0\\ x^{2} + 2x -7 +7&= 0 +7\\ x^{2} + 2x &=7\\ x^{2} + 2x + 1 &=7 + 1\\ x^{2} + 2x + 1 &=8\\ (x + 1)^{2}&= 8\\ x + 1 &= \pm\sqrt{8}\\ x + 1 -1&=-1 \pm\sqrt{8}\\ x&= -1\pm\sqrt{8} \end{align}$

The solution is x&= -1\pm\sqrt{8} which is -3.8 and 1.8.

Solving Quadratic Equations with the Quadratic Formula

If there is no way to factor a quadratic equation or you simply prefer, you can always use the quadratic formula to determine the value(s) of x.

$ax^{2} + bx + c = 0$

Use the quadratic formula to solve this equation but first use the discriminant to learn about the roots.

$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Using and Understanding the Discriminant

The discriminant is the value under the radical, and it provides valuable information about the roots of an quadratic equation.

Discriminant ${b^{2}-4ac}$

• if > 0 then there are two real roots
• if = 0 there is one root repeated
• if < 0 there are two complex roots

For this problem, the discriminant is greater than zero, so there are two real roots.

$x^{2} -6x -4 =0$

$\begin{align} x&=\frac{6\pm\sqrt{36+16}}{2}\\ x&=\frac{6\pm\sqrt{52}}{2}\\ x &= \frac{6}{2}\pm\frac{\sqrt{52}}{2} x&=3\pm 3.6\\ x&= -0.6, 6.6 \end{align}$

The roots for this equation are -0.6 and 6.6.