Solve 357=2[x^2+4x(2x+4)]

$$357 = 2 [ x ^ { 2 } + 4 x ( 2 x + 4 ) ]$$

Solve for x

$$x=\frac{\sqrt{6682}}{18}-\frac{8}{9}\approx 3.652416737$$

$$x=-\frac{\sqrt{6682}}{18}-\frac{8}{9}\approx -5.430194515$$

Steps Using the Quadratic Formula

Divide both sides by 2.

$$\frac{357}{2}=x^{2}+4x\left(2x+4\right)$$

Use the distributive property to multiply 4x by 2x+4.

$$\frac{357}{2}=x^{2}+8x^{2}+16x$$

Combine x^{2} and 8x^{2} to get 9x^{2}.

$$\frac{357}{2}=9x^{2}+16x$$

Swap sides so that all variable terms are on the left hand side.

$$9x^{2}+16x=\frac{357}{2}$$

Subtract \frac{357}{2} from both sides.

$$9x^{2}+16x-\frac{357}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 16 for b, and -\frac{357}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 9\left(-\frac{357}{2}\right)}}{2\times 9}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 9\left(-\frac{357}{2}\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-16±\sqrt{256-36\left(-\frac{357}{2}\right)}}{2\times 9}$$

Multiply -36 times -\frac{357}{2}.

$$x=\frac{-16±\sqrt{256+6426}}{2\times 9}$$

Add 256 to 6426.

$$x=\frac{-16±\sqrt{6682}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-16±\sqrt{6682}}{18}$$

Now solve the equation x=\frac{-16±\sqrt{6682}}{18} when ± is plus. Add -16 to \sqrt{6682}.

$$x=\frac{\sqrt{6682}-16}{18}$$

Divide -16+\sqrt{6682} by 18.

$$x=\frac{\sqrt{6682}}{18}-\frac{8}{9}$$

Now solve the equation x=\frac{-16±\sqrt{6682}}{18} when ± is minus. Subtract \sqrt{6682} from -16.

$$x=\frac{-\sqrt{6682}-16}{18}$$

Divide -16-\sqrt{6682} by 18.

$$x=-\frac{\sqrt{6682}}{18}-\frac{8}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{6682}}{18}-\frac{8}{9} x=-\frac{\sqrt{6682}}{18}-\frac{8}{9}$$

Steps for Completing the Square

Divide both sides by 2.

$$\frac{357}{2}=x^{2}+4x\left(2x+4\right)$$

Use the distributive property to multiply 4x by 2x+4.

$$\frac{357}{2}=x^{2}+8x^{2}+16x$$

Combine x^{2} and 8x^{2} to get 9x^{2}.

$$\frac{357}{2}=9x^{2}+16x$$

Swap sides so that all variable terms are on the left hand side.

$$9x^{2}+16x=\frac{357}{2}$$

Divide both sides by 9.

$$\frac{9x^{2}+16x}{9}=\frac{\frac{357}{2}}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{16}{9}x=\frac{\frac{357}{2}}{9}$$

Divide \frac{357}{2} by 9.

$$x^{2}+\frac{16}{9}x=\frac{119}{6}$$

Divide \frac{16}{9}, the coefficient of the x term, by 2 to get \frac{8}{9}. Then add the square of \frac{8}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{9}x+\left(\frac{8}{9}\right)^{2}=\frac{119}{6}+\left(\frac{8}{9}\right)^{2}$$

Square \frac{8}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{9}x+\frac{64}{81}=\frac{119}{6}+\frac{64}{81}$$

Add \frac{119}{6} to \frac{64}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{9}x+\frac{64}{81}=\frac{3341}{162}$$

Factor x^{2}+\frac{16}{9}x+\frac{64}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{9}\right)^{2}=\frac{3341}{162}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{9}\right)^{2}}=\sqrt{\frac{3341}{162}}$$

Simplify.

$$x+\frac{8}{9}=\frac{\sqrt{6682}}{18} x+\frac{8}{9}=-\frac{\sqrt{6682}}{18}$$

Subtract \frac{8}{9} from both sides of the equation.

$$x=\frac{\sqrt{6682}}{18}-\frac{8}{9} x=-\frac{\sqrt{6682}}{18}-\frac{8}{9}$$