Solve for y (y^2-17)^2-16(y^2-17)=-64

${(y^{2} - 17)}^{2} - 16 (y^{2} - 17) = - 64$

Simplify each term.

Apply the distributive property.

${(y^{2} - 17)}^{2} - 16 y^{2} - 16 \cdot - 17 = - 64$

Multiply $- 16$ by $- 17$ .

${(y^{2} - 17)}^{2} - 16 y^{2} + 272 = - 64$

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 50 u + 561 = - 64$

$u = y^{2}$

Add $64$ to both sides of the equation.

$u^{2} - 50 u + 561 + 64 = 0$

Add $561$ and $64$ .

$u^{2} - 50 u + 625 = 0$

Factor using the perfect square rule.

Rewrite $625$ as $25^{2}$ .

$u^{2} - 50 u + 25^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$50 u = 2 \cdot u \cdot 25$

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 25 + 25^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 25$ .

${(u - 25)}^{2} = 0$

Set the $u - 25$ equal to $0$ .

$u - 25 = 0$

Add $25$ to both sides of the equation.

$u = 25$

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 25$

Solve the equation for $y$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{25}$

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$y = \pm \sqrt{5^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 5$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = 5$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 5$

The complete solution is the result of both the positive and negative portions of the solution.

$y = 5, - 5$