Solve x^2+4(-4x-6)^2=16

$$x ^ { 2 } + 4 ( - 4 x - 6 ) ^ { 2 } = 16$$

Solve for x

$$x=\frac{8\sqrt{14}-96}{65}\approx -1.016411399$$

$$x=\frac{-8\sqrt{14}-96}{65}\approx -1.937434755$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-4x-6\right)^{2}.

$$x^{2}+4\left(16x^{2}+48x+36\right)=16$$

Use the distributive property to multiply 4 by 16x^{2}+48x+36.

$$x^{2}+64x^{2}+192x+144=16$$

Combine x^{2} and 64x^{2} to get 65x^{2}.

$$65x^{2}+192x+144=16$$

Subtract 16 from both sides.

$$65x^{2}+192x+144-16=0$$

Subtract 16 from 144 to get 128.

$$65x^{2}+192x+128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 65 for a, 192 for b, and 128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-192±\sqrt{192^{2}-4\times 65\times 128}}{2\times 65}$$

Square 192.

$$x=\frac{-192±\sqrt{36864-4\times 65\times 128}}{2\times 65}$$

Multiply -4 times 65.

$$x=\frac{-192±\sqrt{36864-260\times 128}}{2\times 65}$$

Multiply -260 times 128.

$$x=\frac{-192±\sqrt{36864-33280}}{2\times 65}$$

Add 36864 to -33280.

$$x=\frac{-192±\sqrt{3584}}{2\times 65}$$

Take the square root of 3584.

$$x=\frac{-192±16\sqrt{14}}{2\times 65}$$

Multiply 2 times 65.

$$x=\frac{-192±16\sqrt{14}}{130}$$

Now solve the equation x=\frac{-192±16\sqrt{14}}{130} when ± is plus. Add -192 to 16\sqrt{14}.

$$x=\frac{16\sqrt{14}-192}{130}$$

Divide -192+16\sqrt{14} by 130.

$$x=\frac{8\sqrt{14}-96}{65}$$

Now solve the equation x=\frac{-192±16\sqrt{14}}{130} when ± is minus. Subtract 16\sqrt{14} from -192.

$$x=\frac{-16\sqrt{14}-192}{130}$$

Divide -192-16\sqrt{14} by 130.

$$x=\frac{-8\sqrt{14}-96}{65}$$

The equation is now solved.

$$x=\frac{8\sqrt{14}-96}{65} x=\frac{-8\sqrt{14}-96}{65}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-4x-6\right)^{2}.

$$x^{2}+4\left(16x^{2}+48x+36\right)=16$$

Use the distributive property to multiply 4 by 16x^{2}+48x+36.

$$x^{2}+64x^{2}+192x+144=16$$

Combine x^{2} and 64x^{2} to get 65x^{2}.

$$65x^{2}+192x+144=16$$

Subtract 144 from both sides.

$$65x^{2}+192x=16-144$$

Subtract 144 from 16 to get -128.

$$65x^{2}+192x=-128$$

Divide both sides by 65.

$$\frac{65x^{2}+192x}{65}=-\frac{128}{65}$$

Dividing by 65 undoes the multiplication by 65.

$$x^{2}+\frac{192}{65}x=-\frac{128}{65}$$

Divide \frac{192}{65}, the coefficient of the x term, by 2 to get \frac{96}{65}. Then add the square of \frac{96}{65} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{192}{65}x+\left(\frac{96}{65}\right)^{2}=-\frac{128}{65}+\left(\frac{96}{65}\right)^{2}$$

Square \frac{96}{65} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{192}{65}x+\frac{9216}{4225}=-\frac{128}{65}+\frac{9216}{4225}$$

Add -\frac{128}{65} to \frac{9216}{4225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{192}{65}x+\frac{9216}{4225}=\frac{896}{4225}$$

Factor x^{2}+\frac{192}{65}x+\frac{9216}{4225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{96}{65}\right)^{2}=\frac{896}{4225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{96}{65}\right)^{2}}=\sqrt{\frac{896}{4225}}$$

Simplify.

$$x+\frac{96}{65}=\frac{8\sqrt{14}}{65} x+\frac{96}{65}=-\frac{8\sqrt{14}}{65}$$

Subtract \frac{96}{65} from both sides of the equation.

$$x=\frac{8\sqrt{14}-96}{65} x=\frac{-8\sqrt{14}-96}{65}$$