Question

This is Josh’s solution for the equation x^2 - 16x - 29 = 7.

x^2 - 16x - 29 = 7
x^2 - 16x = 36
x^2 - 16x + 64 = 36
(x - 8)^2 = 36

x - 8 = 6
x = 14
x - 8 = -6
x = 2
Where is the mistake made? Be specific and explain why it is a mistake.
Solve the equation correctly. Show your work and explain your steps. If you do not explain or show your work, you will not receive credit.

Answer 1

x = 18 or x = -2

Explanation:

Solve for x:

x^2 - 16 x - 29 = 7

Hint: | Solve the quadratic equation by completing the square.

Add 29 to both sides:

x^2 - 16 x = 36

Hint: | Take one half of the coefficient of x and square it, then add it to both sides.

Add 64 to both sides:

x^2 - 16 x + 64 = 100 <- error

Hint: | Factor the left hand side.

Write the left hand side as a square:

(x - 8)^2 = 100

Hint: | Eliminate the exponent on the left hand side.

Take the square root of both sides:

x - 8 = 10 or x - 8 = -10

Hint: | Look at the first equation: Solve for x.

Add 8 to both sides:

x = 18 or x - 8 = -10

Hint: | Look at the second equation: Solve for x.

Add 8 to both sides:

Answer: x = 18 or x = -2

Answer 2

Explanation:

The given quadratic equation is expressed as

x² - 16x - 29 = 7

Looking at Josh's steps, he was trying to apply the method of completing the square to solve the equation. The first and second steps were correct but the third step was wrong. He failed to add the square of half the coefficient of x to the right hand side of the equation.

To solve the equation, we would add (- 16 /2)² to the left hand side and the right hand side of the equation. It becomes

x² - 16x + (- 16/2)² = 36 + (- 16/2)²

(x - 8)² = 36 + 64 = 100

Taking square root of both sides,

x - 8 = ±10

x = 10 + 8 or x = - 10 + 8

x = 18 or x = - 2