More on the Quadratic Formula

Description

Read this section to review the process of solving quadratic equations by using the quadratic formula.

Solve Quadratic Equations Using the Quadratic Formula

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 of solutions of a quadratic equation
Identify the most appropriate method to use to solve a quadratic equation

BE PREPARED 10.7

Before you get started, take this readiness quiz.

Simplify: $\frac{-20-5}{10}$ .

BE PREPARED 10.8

Simplify: $4+\sqrt{121}$ .

BE PREPARED 10.9

Simplify: $\sqrt{128}$ .

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'. 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 in general to solve a quadratic equation for $x$ . It may be helpful to look at one of the examples at the end of the last section where we solved an equation of the form $a x^{2}+b x+c=0$ as you read through the algebraic steps below, so you see them with numbers as well as 'in general'.

We start with the standard form of a quadratic equation and solve it for $x$ by completing the square.	$a x^{2}+b x+c=0$ $a \neq 0$
Isolate the variable terms on one side.	$a x^{2}+b x=-c$
Make leading coefficient 1, by dividing by $a$ .	$\frac{a x^{2}}{a}+\frac{b}{a} x=-\frac{c}{a}$
Simplify.	$x^{2}+\frac{b}{a} x=-\frac{c}{a}$
To complete the square, find $\left(\frac{1}{2} \cdot \frac{b}{a}\right)^{2}$ and add it to both sides of the equation. $\left(\frac{1}{2} \frac{b}{a}\right)^{2}=\frac{b^{2}}{4 a^{2}}$	$x^{2}+\frac{b}{a} x+\frac{b^{2}}{4 a^{2}}=-\frac{c}{a}+\frac{b^{2}}{4 a^{2}}$
The left side is a perfect square, factor it.	$\left(x+\frac{b}{2 a}\right)^{2}=-\frac{c}{a}+\frac{b^{2}}{4 a^{2}}$
Find the common denominator of the right side and write equivalent fractions with the common denominator.	$\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}}{4 a^{2}}-\frac{c \cdot 4 a}{a \cdot 4 a}$
Simplify.	$\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}}{4 a^{2}}-\frac{4 a c}{4 a^{2}}$
Combine to one fraction.	$\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}-4 a c}{4 a^{2}}$
Use the square root property.	$x+\frac{b}{2 a}=\pm \sqrt{\frac{b^{2}-4 a c}{4 a^{2}}}$
Simplify.	$x+\frac{b}{2 a}=\pm \frac{\sqrt{b^{2}-4 a c}}{2 a}$
Add $-\frac{b}{2 a}$ to both sides of the equation.	$x=-\frac{b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a}$
Combine the terms on the right side.	$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$

Source: Rice University, https://openstax.org/books/elementary-algebra-2e/pages/10-3-solve-quadratic-equations-using-the-quadratic-formula
This work is licensed under a Creative Commons Attribution 4.0 License.

Quadratic Formula

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

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

To use the Quadratic Formula, we substitute the values of $a$ , $b$ and $c$ into the expression on the right side of the formula. Then, we do all the math to simplify the expression. The result gives the solution(s) to the quadratic equation.

EXAMPLE 10.28

How to Solve a Quadratic Equation Using the Quadratic Formula

Solve $2 x^{2}+9 x-5=0$ by using the Quadratic Formula.

Solution

TRY IT 10.55

Solve $3 y^{2}-5 y+2=0$ by using the Quadratic Formula.

TRY IT 10.56

Solve $4 z^{2}+2 z-6=0$ by using the Quadratic Formula.

HOW TO

Solve a quadratic equation using the Quadratic Formula.

Step 1: Write the Quadratic Formula in standard form. Identify the $a$ , $b$ and $c$ values.

Step 2: Write the Quadratic Formula. Then substitute in the values of $a$ , $b$ and $c$ values.

Step 3: Simplify.

Step 4: 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 10.29

Solve $x^{2}-6 x+5=0$ by using the Quadratic Formula.

Solution


This equation is in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$
Simplify.
Rewrite to show two solutions.
Simplify.
Check.

TRY IT 10.57

Solve $a^{2}-2 a-15=0$ by using the Quadratic Formula.

TRY IT 10.58

Solve $b^{2}+10 b+24=0$ by using the Quadratic Formula.

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 10.30

Solve $4 y^{2}-5 y-3=0$ by using the Quadratic Formula.

Solution

We can use the Quadratic Formula to solve for the variable in a quadratic equation, whether or not it is named 'x'.


This equation is in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$
Simplify.
Rewrite to show two solutions.
Check. We leave the check to you.

TRY IT 10.59

Solve $2 p^{2}+8 p+5=0$ by using the Quadratic Formula.

TRY IT 10.60

Solve $5 q^{2}-11 q+3=0$ by using the Quadratic Formula.

EXAMPLE 10.31

Solve $2 x^{2}+10 x+11=0$ by using the Quadratic Formula.

Solution


This equation is in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$
Simplify.
Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check. We leave the check to you.

TRY IT 10.61

Solve $3 m^{2}+12 m+7=0$ by using the Quadratic Formula.

TRY IT 10.62

Solve $5 n^{2}+4 n-4=0$ by using the Quadratic Formula.

We cannot take the square root of a negative number. So, when we substitute $a$ , $b$ , and $c$ into the Quadratic Formula, if the quantity inside the radical is negative, the quadratic equation has no real solution. We will see this in the next example.

EXAMPLE 10.32

Solve $3 p^{2}+2 p+9=0$ by using the Quadratic Formula.

Solution.

This equation is in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$ .
Simplify.
Simplify the radical.
We cannot take the square root of a negative number.	There is no real solution.

TRY IT 10.63

Solve $4 a^{2}-3 a+8=0$ by using the Quadratic Formula.

TRY IT 10.64

Solve $5 b^{2}+2 b+4=0$ by using the Quadratic Formula.

The quadratic equations we have solved so far in this section were all 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 10.33

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

Solution


Distribute to get the equation in standard form.
This equation is now in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$
Simplify.
Simplify inside the radical.
Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check. We leave the check to you.

TRY IT 10.65

Solve $x(x+2)-5=0$ by using the Quadratic Formula.

TRY IT 10.66

Solve $y(3y−1)−2=0$ by using the Quadratic Formula.

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 10.34

Solve $\frac{1}{2} u^{2}+\frac{2}{3} u=\frac{1}{3}$ by using the Quadratic Formula.

Solution


Multiply both sides by the LCD, 6, to clear the fractions.
Multiply.
Subtract 2 to get the equation in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$
Simplify.
Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check. We leave the check to you.

TRY IT 10.67

Solve $\frac{1}{4} c^{2}-\frac{1}{3} c=\frac{1}{12}$ by using the Quadratic Formula.

TRY IT 10.68

Solve $\frac{1}{9} d^{2}-\frac{1}{2} d=-\frac{1}{2}$ by using the Quadratic Formula.

Think about the equation $(x-3)^{2}=0$ . We know from the Zero Products Principle 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 with a perfect square also gives just one solution.

EXAMPLE 10.35

Solve $4 x^{2}-20 x=-25$ by using the Quadratic Formula.

Solution


Add 25 to get the equation in standard form.
Identify the $a$ , $b$ , $c$ values.
Write the Quadratic Formula.
Then substitute in the values of $a$ , $b$ , $c$
Simplify.
Simplify the radical.
Simplify the fraction.
Check. We leave the check to you.