Solve (4x+4x+x^2)*2=312

$$(4x+4x+ { x }^{ 2 } ) \times 2=312$$

Solve for x

$$x=2\sqrt{43}-4\approx 9.114877049$$

$$x=-2\sqrt{43}-4\approx -17.114877049$$

Steps Using the Quadratic Formula

Divide both sides by 2.

$$4x+4x+x^{2}=\frac{312}{2}$$

Divide 312 by 2 to get 156.

$$4x+4x+x^{2}=156$$

Combine 4x and 4x to get 8x.

$$8x+x^{2}=156$$

Subtract 156 from both sides.

$$8x+x^{2}-156=0$$

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.

$$x^{2}+8x-156=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -156 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-156\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-156\right)}}{2}$$

Multiply -4 times -156.

$$x=\frac{-8±\sqrt{64+624}}{2}$$

Add 64 to 624.

$$x=\frac{-8±\sqrt{688}}{2}$$

Take the square root of 688.

$$x=\frac{-8±4\sqrt{43}}{2}$$

Now solve the equation x=\frac{-8±4\sqrt{43}}{2} when ± is plus. Add -8 to 4\sqrt{43}.

$$x=\frac{4\sqrt{43}-8}{2}$$

Divide -8+4\sqrt{43} by 2.

$$x=2\sqrt{43}-4$$

Now solve the equation x=\frac{-8±4\sqrt{43}}{2} when ± is minus. Subtract 4\sqrt{43} from -8.

$$x=\frac{-4\sqrt{43}-8}{2}$$

Divide -8-4\sqrt{43} by 2.

$$x=-2\sqrt{43}-4$$

The equation is now solved.

$$x=2\sqrt{43}-4 x=-2\sqrt{43}-4$$

Steps for Completing the Square

Divide both sides by 2.

$$4x+4x+x^{2}=\frac{312}{2}$$

Divide 312 by 2 to get 156.

$$4x+4x+x^{2}=156$$

Combine 4x and 4x to get 8x.

$$8x+x^{2}=156$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+8x=156$$

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.

$$x^{2}+8x+4^{2}=156+4^{2}$$

Square 4.

$$x^{2}+8x+16=156+16$$

Add 156 to 16.

$$x^{2}+8x+16=172$$

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}.

$$\left(x+4\right)^{2}=172$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{172}$$

Simplify.

$$x+4=2\sqrt{43} x+4=-2\sqrt{43}$$

Subtract 4 from both sides of the equation.

$$x=2\sqrt{43}-4 x=-2\sqrt{43}-4$$