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Algebra 1

9.6.3 The Quadratic Formula

Algebra 19.6.3 The Quadratic Formula

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Activity

Here is a formula called the quadratic formula.

x=b±b24ac2ax=b±b24ac2a

The formula can be used to find the solutions to any quadratic equation in the form of ax2+bx+c=0ax2+bx+c=0, where aa, bb, and cc are numbers and aa is not 0.

This example shows how it is used to solve x28x+15=0x28x+15=0, in which a=a=1, b=8b=8, and c=15c=15.

Step 1 - State the original equation.

x28x+15=0x28x+15=0

Step 2 - Identify the values of aa, bb, and cc.

a=1a=1, b=8b=8, and c=15c=15

Step 3 - Substitute the values of aa, bb, and cc into the formula.

x=(8)±(8)2(4)(1)(15)2(1)x=(8)±(8)2(4)(1)(15)2(1)

Step 4 - Evaluate each part of the expression.

x=8±64602x=8±64602

x=8±42x=8±42

x=8±22x=8±22

Step 5 - Write each equation separately and solve.

x=8+22x=8+22 and x=822x=822

Step 6 - Simplify.

x=102x=102 and x=62x=62

Step 7 - Find the solutions.

x=5x=5 and x=3x=3

Here are some quadratic equations and their solutions. Use the quadratic formula to show that the solutions are correct.

1.

x2+4x5=0x2+4x5=0. The solutions are x=5x=5 and x=1x=1.

2.

x2+7x+12=0x2+7x+12=0. The solutions are x=3x=3 and x=4x=4.

3.

x2+10x+18=0x2+10x+18=0. The solution is x=5±7x=5±7.

4.

x28x+11=0x28x+11=0. The solution is x=4±5x=4±5.

5.

9x26x+1=09x26x+1=0. The solution is x=13x=13.

6.

6x2+9x15=06x2+9x15=0. The solutions are x=52x=52 and x=1x=1.

Video: Solving Quadratics With the Quadratic Formula

Watch the following video to learn more about solving quadratics with the quadratic formula.

Self Check

Solve 2 x 2 6 x + 3 = 0 using the quadratic formula.
  1. x = 6 ± 12 4
  2. x = 6 ± 60 4
  3. x = 6 ± 12 4
  4. x = 6 ± 12 4

Additional Resources

Solving With the Quadratic Formula

Quadratic Formula

The solutions to a quadratic equation of the form ax2+bx+cax2+bx+c, where a0a0, are given by the formula:

x=b±b24ac2ax=b±b24ac2a

To use the quadratic formula, we substitute the values of aa, bb, and cc 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 formula is an equation. Make sure you use both sides of the equation.

How to Solve a Quadratic Equation Using the Quadratic Formula

Solve by using the quadratic formula: 2x2+9x5=02x2+9x5=0.

Step 1 - Write the quadratic equation in standard form.

This equation is in standard form.

ax2+bx+c=0ax2+bx+c=0

2x2+9x5=02x2+9x5=0

Step 2 - Identify the aa, bb, and cc values.

a=2a=2, b=9b=9, c=5c=5

Step 3 - Write the quadratic formula.

x=b±b24ac2ax=b±b24ac2a

Step 4 - Then substitute in the values of aa, bb, and cc.

a=2a=2, b=9b=9, c=5c=5

x=9±924·2·(5)2·2x=9±924·2·(5)2·2

Step 5 - Simplify the fraction and solve for xx.

x=9±81(40)4x=9±81(40)4

x=9±1214x=9±1214

x=9+114x=9+114

x=9114x=9114

x=24x=24

x=204x=204

x=12x=12

x=5x=5

Step 6 - Check the solutions. Put each answer in the original equation to check. Substitute x=12x=12.

2(12)2+9·125 = ? 02(14)+9·125 = ? 024+925 = ? 012+925 = ? 01025 = ? 055 = ? 00=0 ✔2(12)2+9·125 = ? 02(14)+9·125 = ? 024+925 = ? 012+925 = ? 01025 = ? 055 = ? 00=0 ✔

Substitute x=5x=5.

2(5)2+9(5)5 = ? 02(25)455 = ? 050455 = ? 00=0 ✔2(5)2+9(5)5 = ? 02(25)455 = ? 050455 = ? 00=0 ✔

Try it

Try It: Solving With the Quadratic Formula

Solve 2x2+10x+11=02x2+10x+11=0 using the quadratic formula.

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