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Question Solve the quadratic equation 2(−5x−6)2=42(-5x-6)^2=4 by extracting the roots.

Studdy Solution

STEP 1

1. The equation 2(−5x−6)2=42(-5x-6)^2=4 can be solved by extracting roots, which involves taking the square root of both sides of the equation.
2. The square root of a squared expression yields two possible solutions, one positive and one negative.
3. After extracting the roots, further algebraic manipulations may be required to isolate xx and find its value(s).

STEP 2

1. Divide both sides of the equation by 2 to isolate the squared term.
2. Take the square root of both sides of the equation to eliminate the exponent.
3. Solve for xx by isolating it on one side of the equation.

STEP 3

Divide both sides of the equation by 2 to isolate the squared term.
2(−5x−6)22=42 \frac{2(-5x-6)^2}{2} = \frac{4}{2}

STEP 4

Simplify the equation.
(−5x−6)2=2 (-5x-6)^2 = 2

STEP 5

Take the square root of both sides of the equation to eliminate the exponent.
(−5x−6)2=±2 \sqrt{(-5x-6)^2} = \pm\sqrt{2}

STEP 6

Recognize that the square root of a squared expression yields the absolute value of the expression, which gives two possible solutions.
−5x−6=2or−5x−6=−2 -5x-6 = \sqrt{2} \quad \text{or} \quad -5x-6 = -\sqrt{2}

STEP 7

Solve for xx in the first equation by isolating it on one side.
−5x=2+6 -5x = \sqrt{2} + 6

STEP 8

Divide both sides of the first equation by -5 to find the first solution for xx.
x=2+6−5 x = \frac{\sqrt{2} + 6}{-5}

STEP 9

Solve for xx in the second equation by isolating it on one side.
−5x=−2+6 -5x = -\sqrt{2} + 6

STEP 10

Divide both sides of the second equation by -5 to find the second solution for xx.
x=−2+6−5 x = \frac{-\sqrt{2} + 6}{-5}

STEP 11

Combine the two solutions for xx into a single statement.
x=2+6−5orx=−2+6−5 x = \frac{\sqrt{2} + 6}{-5} \quad \text{or} \quad x = \frac{-\sqrt{2} + 6}{-5}

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