Solve (x^2-16)^2-10(x^2-16)+9=0

$$( x ^ { 2 } - 16 ) ^ { 2 } - 10 ( x ^ { 2 } - 16 ) + 9 = 0$$

Solve for x

$$x=-5$$

$$x=5$$

$$x=\sqrt{17}\approx 4.123105626$$

$$x=-\sqrt{17}\approx -4.123105626$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-16\right)^{2}.

$$\left(x^{2}\right)^{2}-32x^{2}+256-10\left(x^{2}-16\right)+9=0$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-32x^{2}+256-10\left(x^{2}-16\right)+9=0$$

Use the distributive property to multiply -10 by x^{2}-16.

$$x^{4}-32x^{2}+256-10x^{2}+160+9=0$$

Combine -32x^{2} and -10x^{2} to get -42x^{2}.

$$x^{4}-42x^{2}+256+160+9=0$$

Add 256 and 160 to get 416.

$$x^{4}-42x^{2}+416+9=0$$

Add 416 and 9 to get 425.

$$x^{4}-42x^{2}+425=0$$

Substitute t for x^{2}.

$$t^{2}-42t+425=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -42 for b, and 425 for c in the quadratic formula.

$$t=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 1\times 425}}{2}$$

Do the calculations.

$$t=\frac{42±8}{2}$$

Solve the equation t=\frac{42±8}{2} when ± is plus and when ± is minus.

$$t=25 t=17$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=5 x=-5 x=\sqrt{17} x=-\sqrt{17}$$