Solve 4x^2=x^2+64-16x+x^2

$$4 { x }^{ 2 } = { x }^{ 2 } +64-16x+ { x }^{ 2 }$$

Solve for x

$$x=4\sqrt{3}-4\approx 2.92820323$$

$$x=-4\sqrt{3}-4\approx -10.92820323$$

Steps Using the Quadratic Formula

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

$$4x^{2}=2x^{2}+64-16x$$

Subtract 2x^{2} from both sides.

$$4x^{2}-2x^{2}=64-16x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}=64-16x$$

Subtract 64 from both sides.

$$2x^{2}-64=-16x$$

Add 16x to both sides.

$$2x^{2}-64+16x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}+16x-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 16 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\left(-64\right)}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\left(-64\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\left(-64\right)}}{2\times 2}$$

Multiply -8 times -64.

$$x=\frac{-16±\sqrt{256+512}}{2\times 2}$$

Add 256 to 512.

$$x=\frac{-16±\sqrt{768}}{2\times 2}$$

Take the square root of 768.

$$x=\frac{-16±16\sqrt{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-16±16\sqrt{3}}{4}$$

Now solve the equation x=\frac{-16±16\sqrt{3}}{4} when ± is plus. Add -16 to 16\sqrt{3}.

$$x=\frac{16\sqrt{3}-16}{4}$$

Divide -16+16\sqrt{3} by 4.

$$x=4\sqrt{3}-4$$

Now solve the equation x=\frac{-16±16\sqrt{3}}{4} when ± is minus. Subtract 16\sqrt{3} from -16.

$$x=\frac{-16\sqrt{3}-16}{4}$$

Divide -16-16\sqrt{3} by 4.

$$x=-4\sqrt{3}-4$$

The equation is now solved.

$$x=4\sqrt{3}-4 x=-4\sqrt{3}-4$$

Steps for Completing the Square

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

$$4x^{2}=2x^{2}+64-16x$$

Subtract 2x^{2} from both sides.

$$4x^{2}-2x^{2}=64-16x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}=64-16x$$

Add 16x to both sides.

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

Divide both sides by 2.

$$\frac{2x^{2}+16x}{2}=\frac{64}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{16}{2}x=\frac{64}{2}$$

Divide 16 by 2.

$$x^{2}+8x=\frac{64}{2}$$

Divide 64 by 2.

$$x^{2}+8x=32$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=32+4^{2}$$

Square 4.

$$x^{2}+8x+16=32+16$$

Add 32 to 16.

$$x^{2}+8x+16=48$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+4=4\sqrt{3} x+4=-4\sqrt{3}$$

Subtract 4 from both sides of the equation.

$$x=4\sqrt{3}-4 x=-4\sqrt{3}-4$$