3.04 Solving Quadratic Equations by the Quadratic Formula

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The Formula

The Quadratic Formula

The quadratic formula can be used to solve any quadratic equation. To solve the equation, you will always want your equation in standard form, ax² + bx + c = 0. Notice the right side of the equation must be zero. a, b, and c each represent real numbers.

The quadratic formula is: $x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a$

Identifying Values

It is important to identify the values for a, b, and c in order to use the quadratic formula.

Recall that a, b, and c represent the coefficients and constant of a quadratic equation.

For example, 4x² − 6x + 3 = 0

a = 4
b = −6
c = 3

Example #1

Identify the values for a, b, and c using the quadratic equation 2x² + 5x − 5 = 0.

a =
b =
c =

Example #2

Identify the values for a, b, and c using the quadratic equation −8x² − 9x − 1 = 0.

a =
b =
c =

Substituting in the Formula

When substituting the values for a, b, and c into the quadratic formula, it is suggested to use parentheses so you will substitute the correct values.

Suppose you had the expression −3x² − 2x + 3. a = −3, b = −2, and c = 3. Substitute these values into the quadratic formula:

$x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a$

$x equals the fraction with numerator negative negative 2 plus or minus the square root of open paren negative 2 close paren squared minus 4 times negative 3 times 3 and denominator 2 times negative 3$

Hint: When substituting the negative value, remember to write the sign inside the parentheses. In this example, b = −2. When you substitute for the first b of the formula, don't forget the negative. So, you will have −(−2) for the first substitution.

Example #3

Practice substituting into the quadratic formula using the quadratic equation: 5x² − 7x + 3 = 0.

$x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a$

$x equals the fraction with the numerator negative blank plus or minus the square root of blank squared minus four times blank times blank and denominator two times blank$

Example #4

Practice substituting into the quadratic formula using the quadratic equation: −9x² + 4x − 6 = 0.

$x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a$

$x equals the fraction with the numerator negative blank plus or minus the square root of blank squared minus four times blank times blank and denominator two times blank$

Simplify

Simplifying the Quadratic Formula

Now that you know how to substitute into the quadratic formula, it is time to learn how to simplify the formula.

Example #5

Open Solve Quadratic Equations Using the Quadratic Formula. in a new tab

Example #6

Open Solve Quadratic Equations Using the Quadratic Formula. in a new tab

Example #7

Solve the quadratic function x² + 7x + 3 = 0 using the quadratic formula.

a =
b =
c =

Using the values a = 1, b = 7, and c = 3, fill-in the formula.

$x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a$

$x equals the fraction with numerator negative 7 plus or minus the square root of 7 squared minus 4 times 1 times 3 and denominator 2 times 1$

Simplify the first b term and the squared term. You can also simplify the denominator.

$x equals the fraction with numerator negative 7 plus or minus the square root of 49 minus 4 times 1 times 3 and denominator 2$

Simplify the multiplication under the radical.

$x equals the fraction with numerator negative 7 plus or minus the square root of 49 minus 12 and denominator 2$

Simplify the subtraction under the radical.

$x equals the fraction with numerator negative 7 plus or minus the square root of 37 and denominator 2$

Simplify the square root by finding a decimal.

$x equals the fraction with numerator negative 7 plus or minus the square root of 6.08 and denominator 2$

Separate the two parts of the equation and solve. Simplify the numerators.

Part 1:

$x equals the fraction with numerator negative 7 plus 6.08 and denominator 2$
$x equals the fraction with numerator negative 0.92 and denominator 2$
x = −0.46

Part 2:

$x equals the fraction with numerator negative 7 minus 6.08 and denominator 2$
x = −6.54

The two solutions are x = −0.46 and x = −6.54 for the quadratic equation.

Remember that you can easily check your answer by substituting these values into the original question. This is always a good step to do so you will know your answer is correct.

Common Mistakes

As you probably have noticed, the quadratic formula is quite an intricate formula. There are a few common mistakes students tend to make when working with this formula.

Let's explore these potential mistakes to help in your solving.

Common Mistakes #1

The first common mistake is not properly substituting for the b value. Suppose you have 6x² − 3x + 10 = 0. In this equation a = 6, b = −3, and c = 10.

Correct Substitution

$x equals the fraction with numerator negative negative 3 plus or minus the square root of open paren negative 3 close paren squared minus 4 times 6 times 10 and denominator 2 times 6$

Incorrect Substitution

$x equals the fraction with numerator negative 3 plus or minus the square root of negative 3 squared minus 4 times 6 times 10 and denominator 2 times 6$

Common Mistakes #2

Another common mistake is not dividing the entire expression by 2a. Suppose you have 6x² − 3x + 10 = 0. In this equation a = 6, b = −3, and c = 10.

Correct Substitution

$x equals the fraction with numerator negative negative 3 plus or minus the square root of open paren negative 3 close paren squared minus 4 times 6 times 10 and denominator 2 times 6$

Incorrect Substitution

$x equals negative negative 3 plus or minus the fraction with numerator the square root of open paren negative 3 close paren squared minus 4 times 6 times 10 and denominator 2 times 6$

Common Mistakes #3

Another common mistake is not correctly squaring the b term under the radical. Suppose you have 6x² − 3x + 10 = 0. In this equation a = 6, b = −3, and c = 10.

Correct Substitution

$x equals the fraction with numerator negative negative 3 plus or minus the square root of open paren negative 3 close paren squared minus 4 times 6 times 10 and denominator 2 times 6$

$x equals the fraction with numerator 3 plus or minus the square root of 9 minus 4 times 6 times 10 and denominator 2 times 6$

Incorrect Substitution

$x equals the fraction with numerator negative negative 3 plus or minus the square root of negative 3 squared minus 4 times 6 times 10 and denominator 2 times 6$

$x equals the fraction with numerator 3 plus or minus the square root of negative 9 minus 4 times 6 times 10 and denominator 2 times 6$

Example #8

Using the quadratic equation −4x² − 7x + 3 = 0, which of the following is properly substituted and squared?

$Box A. Two equations. First equation: x equals the fraction with numerator negative negative 7 plus or minus the square root of 7 squared minus 4 times negative 4 times 3 and denominator 2 times negative 4. Second equation: x equals the fraction with numerator 7 plus or minus the square root of 49 plus 48 and denominator negative 8$
$Box B. Two equations. First equation: x equals the fraction with numerator 7 plus or minus the square root of 7 squared minus 4 times negative 4 times 3 and denominator 2 times negative 4. Second equation: x equals the fraction with numerator 7 plus or minus the square root of 49 plus 48 and denominator negative 8$
$Box C. Two equations. First equation: x equals the fraction with numerator negative negative 7 plus or minus the square root of negative 7 squared minus 4 times negative 4 times 3 and denominator 2 times negative 4. Second equation: x equals the fraction with numerator 7 plus or minus the square root of 49 plus 48 and denominator negative 8$
$Box C. Two equations. First equation: x equals the fraction with numerator negative 7 plus or minus the square root of negative 7 squared minus 4 times negative 4 times 3 and denominator 2 times negative 4. Second equation: x equals the fraction with numerator 7 plus or minus the square root of 49 plus 48 and denominator negative 8$

Answer: $Box C. Two equations. First equation: x equals the fraction with numerator negative negative 7 plus or minus the square root of negative 7 squared minus 4 times negative 4 times 3 and denominator 2 times negative 4. Second equation: x equals the fraction with numerator 7 plus or minus the square root of 49 plus 48 and denominator negative 8$

Complex Solutions

Example #9

Solve 2x² + 5x + 5 = 1 by using the quadratic formula.

Step 1:

Get everything on the same side. Subtract 1 from both sides.

Step 2:

Identify the a, b, and c and plug into the formula.

a =
b =
c =

$x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a. c and denominator 2 a.$

$x equals the fraction with numerator negative 5 plus or minus the square root of 5 squared minus 4 times 2 times 4 and denominator 2 times 2$

Step 3:

Simplify:
$x equals the fraction with numerator negative 5 plus or minus the square root of 25 minus 32 and denominator 4$
$x equals the fraction with numerator negative 5 plus or minus the square root of negative 7 and denominator 4$

Remember, when you take the square root of a negative number, you get an i.

$x equals the fraction with numerator negative 5 plus or minus i the square root of 7 and denominator 4$

We will not have decimal answers since there is an i in the answer. Remember, if we have a complex solutions, there will not be any x-intercepts on the graph. The graph will never cross the x-axis.

Example #10

Solve −11x² + 6x − 1 = 2 by using the quadratic formula.

Step 1:

Get everything on the same side. Subtract 2 from both sides.

Step 2:

Identify the a, b, and c and plug into the formula.

a =
b =
c =

$x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a$

$x equals the fraction with numerator negative 6 plus or minus 6 squared minus 4 times negative 11 times negative 3 and denominator 2 times negative 11$

Step 3:

Simplify:
$x equals the fraction with numerator negative 6 plus or minus the square root of open bracket 36 minus 132 close bracket and denominator the square root of negative 22$
$x equals the fraction with numerator negative 6 plus or minus the square root of negative 96 and denominator negative 22$

Remember, when you take the square root of a negative number, you get an i.

$x equals the fraction with numerator negative 6 plus or minus i times the square root of 96 and denominator negative 22$

In this problem, we can break down and simplify the square root of 96.

the square root of 96 equals the square root of 6 times 16 equals 4 the square root of 6

$x equals the fraction with numerator negative 6 plus or minus 4 i the square root of 6 and denominator 22$

If all numbers (not including the number in the radical) can reduce by a number, then we can reduce. If only 1 number on top can reduce with the denominator, then you cannot reduce the fraction! −6, 4, and 22 can reduce by 2.

$x equals the fraction with numerator negative 3 plus or minus 2 i the square root of 6 and denominator negative 11$

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