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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 9: Q. 530 (page 988)

Use Completing the Square to solve the quadratic equation $a^{2} - 8 a + 7 = 23$

Short Answer

Expert verified

Using the Completing the Square method, the solution to the quadratic equation is given by:

$a = 4 卤 4 \sqrt{2}$

Step by step solution

01

Step 1. Given information.

We have:

$a^{2} - 8 a + 7 = 23$

02

Step 2. Find the perfect square.

Divide the coefficient of the middle term by $2$ the square it to get the value of the term to be added to get the perfect square.

${(\frac{- 8}{2})}^{2} = 16$

03

Step 3. Add the perfect square to the both sides of the equation.

Add $16$ on both sides of the equation:

$a^{2} - 8 a + 7 + 16 = 23 + 16$

Using identity ${(a - b)}^{2} = a^{2} - 2 a b + b^{2},$ it follows:

${(a - 4)}^{2} + 7 = 23 + 16 {(a - 4)}^{2} + 7 = 49$

${(a - 4)}^{2} = 49 - 7 {(a - 4)}^{2} = 42$

04

Step 4. Solve for a.

Square root on both sides:

$\sqrt{{(a - 4)}^{2}} = \sqrt{42} a - 4 = 卤 4 \sqrt{2} a = 4 卤 4 \sqrt{2}$

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