Math

Question Solve the quadratic equation (x28)22(x28)=1\left(x^{2}-8\right)^{2}-2\left(x^{2}-8\right)=-1 and provide all solutions.

Studdy Solution

STEP 1

1. The equation (x28)22(x28)=1\left(x^{2}-8\right)^{2}-2\left(x^{2}-8\right)=-1 is a quadratic equation in terms of x28x^2 - 8.
2. The solutions for x28x^2 - 8 can be found using the quadratic formula or by factoring.
3. Once the solutions for x28x^2 - 8 are found, we can solve for xx by taking the square root of both sides of the equation.

STEP 2

1. Recognize the equation as a quadratic in terms of x28x^2 - 8.
2. Solve the quadratic equation for x28x^2 - 8.
3. Solve for xx by taking the square root of the solutions for x28x^2 - 8.
4. List all possible solutions for xx.

STEP 3

Recognize the given equation as a quadratic in terms of x28x^2 - 8.
Let y=x28y = x^2 - 8. Then the equation becomes:
y22y=1 y^2 - 2y = -1

STEP 4

Rewrite the equation in standard quadratic form.
y22y+1=0 y^2 - 2y + 1 = 0

STEP 5

Factor the quadratic equation.
(y1)2=0 (y - 1)^2 = 0

STEP 6

Find the solution for yy.
Since (y1)2=0(y - 1)^2 = 0, we have y1=0y - 1 = 0 which implies y=1y = 1.

STEP 7

Substitute back x28x^2 - 8 for yy and solve for xx.
x28=1 x^2 - 8 = 1

STEP 8

Add 8 to both sides of the equation to isolate x2x^2.
x2=9 x^2 = 9

STEP 9

Take the square root of both sides of the equation to solve for xx.
x=±9 x = \pm\sqrt{9}

STEP 10

Calculate the square root of 9.
x=±3 x = \pm 3

STEP 11

List all possible solutions for xx.
The solutions for xx are x=3x = 3 and x=3x = -3.
The solutions are 3,33, -3.

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