Solve 4(x^2-1)^2-9=0

$$4 ( x ^ { 2 } - 1 ) ^ { 2 } - 9 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{2}i}{2}\approx 0.707106781i$$

$$x=-\frac{\sqrt{2}i}{2}\approx -0-0.707106781i$$

$$x = -\frac{\sqrt{10}}{2} \approx -1.58113883$$

$$x = \frac{\sqrt{10}}{2} \approx 1.58113883$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-1\right)^{2}.

$$4\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)-9=0$$

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

$$4\left(x^{4}-2x^{2}+1\right)-9=0$$

Use the distributive property to multiply 4 by x^{4}-2x^{2}+1.

$$4x^{4}-8x^{2}+4-9=0$$

Subtract 9 from 4 to get -5.

$$4x^{4}-8x^{2}-5=0$$

Substitute t for x^{2}.

$$4t^{2}-8t-5=0$$

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 4 for a, -8 for b, and -5 for c in the quadratic formula.

$$t=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\left(-5\right)}}{2\times 4}$$

Do the calculations.

$$t=\frac{8±12}{8}$$

Solve the equation t=\frac{8±12}{8} when ± is plus and when ± is minus.

$$t=\frac{5}{2} t=-\frac{1}{2}$$

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

$$x=-\frac{\sqrt{10}}{2} x=\frac{\sqrt{10}}{2} x=-\frac{\sqrt{2}i}{2} x=\frac{\sqrt{2}i}{2}$$

Solve for x

$$x = -\frac{\sqrt{10}}{2} \approx -1.58113883$$

$$x = \frac{\sqrt{10}}{2} \approx 1.58113883$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-1\right)^{2}.

$$4\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)-9=0$$

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

$$4\left(x^{4}-2x^{2}+1\right)-9=0$$

Use the distributive property to multiply 4 by x^{4}-2x^{2}+1.

$$4x^{4}-8x^{2}+4-9=0$$

Subtract 9 from 4 to get -5.

$$4x^{4}-8x^{2}-5=0$$

Substitute t for x^{2}.

$$4t^{2}-8t-5=0$$

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 4 for a, -8 for b, and -5 for c in the quadratic formula.

$$t=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\left(-5\right)}}{2\times 4}$$

Do the calculations.

$$t=\frac{8±12}{8}$$

Solve the equation t=\frac{8±12}{8} when ± is plus and when ± is minus.

$$t=\frac{5}{2} t=-\frac{1}{2}$$

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

$$x=\frac{\sqrt{10}}{2} x=-\frac{\sqrt{10}}{2}$$