Question

solve the equation x^-2x-11=0 by completing the square. Give your answers correct to 2 decimal places.

Answer 1

Explanation:

I assume the real equation is

x² - 2x - 11 = 0

to complete the square we need to look first on the x² and x terms.

to complete the square we have to bring this into a form

(x + a)² = b

then we can pull the square root on both sides and get

x + a = sqrt(b)

x = sqrt(b) - a

so,

(x + a)² = x² + 2ax + a²

now compare this to our given equation.

x² check.

-2x = 2ax

therefore, a = -1

b is the remaining constant :

(x - 1)² - b = 0

x² - 2x + 1 - b = 0

when comparing to our original equation we see

1 - b = -11

b = 12

so,

x² - 2x - 11 = (x - 1)² - 12 = 0

(x - 1)² = 12

x - 1 = sqrt(12) = sqrt(4×3) = 2×sqrt(3)

x = 2×sqrt(3) + 1 = 4.464101615... ≈ 4.46

Answer 2

• x = -2.46
• x = 4.46

Explanation:

You want the solution to x² -2x -11 = 0 by completing the square.

Completing the square

We can rewrite the quadratic as ...

x² -2x = 11

Adding the square of half the x-coefficient completes the square

x² -2x +(-2/2)² = 11 +(-2/2)²

(x -1)² = 12

Solutions

Taking the square root gives ...

x -1 = ±√12

x = 1 ±2√3

x ≈ {-2.46, 4.46}

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