Solve 0=6x^2+84x+288

$$0 = 6 x ^ { 2 } + 84 x + 288$$

Solve for x

$$x=-8$$

$$x=-6$$

Steps Using the Quadratic Formula

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

$$6x^{2}+84x+288=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 84 for b, and 288 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-84±\sqrt{84^{2}-4\times 6\times 288}}{2\times 6}$$

Square 84.

$$x=\frac{-84±\sqrt{7056-4\times 6\times 288}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-84±\sqrt{7056-24\times 288}}{2\times 6}$$

Multiply -24 times 288.

$$x=\frac{-84±\sqrt{7056-6912}}{2\times 6}$$

Add 7056 to -6912.

$$x=\frac{-84±\sqrt{144}}{2\times 6}$$

Take the square root of 144.

$$x=\frac{-84±12}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-84±12}{12}$$

Now solve the equation x=\frac{-84±12}{12} when ± is plus. Add -84 to 12.

$$x=-\frac{72}{12}$$

Divide -72 by 12.

$$x=-6$$

Now solve the equation x=\frac{-84±12}{12} when ± is minus. Subtract 12 from -84.

$$x=-\frac{96}{12}$$

Divide -96 by 12.

$$x=-8$$

The equation is now solved.

$$x=-6 x=-8$$

Steps for Completing the Square

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

$$6x^{2}+84x+288=0$$

Subtract 288 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}+84x=-288$$

Divide both sides by 6.

$$\frac{6x^{2}+84x}{6}=-\frac{288}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{84}{6}x=-\frac{288}{6}$$

Divide 84 by 6.

$$x^{2}+14x=-\frac{288}{6}$$

Divide -288 by 6.

$$x^{2}+14x=-48$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=-48+7^{2}$$

Square 7.

$$x^{2}+14x+49=-48+49$$

Add -48 to 49.

$$x^{2}+14x+49=1$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+7=1 x+7=-1$$

Subtract 7 from both sides of the equation.

$$x=-6 x=-8$$