Solve 6x^2+84x+280=-8

$$6 { x }^{ 2 } +84x+280=-8$$

Solve for x

$$x=-8$$

$$x=-6$$

Steps Using the Quadratic Formula

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.

$$6x^{2}+84x+280=-8$$

Add 8 to both sides of the equation.

$$6x^{2}+84x+280-\left(-8\right)=-8-\left(-8\right)$$

Subtracting -8 from itself leaves 0.

$$6x^{2}+84x+280-\left(-8\right)=0$$

Subtract -8 from 280.

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

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

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.

$$6x^{2}+84x+280=-8$$

Subtract 280 from both sides of the equation.

$$6x^{2}+84x+280-280=-8-280$$

Subtracting 280 from itself leaves 0.

$$6x^{2}+84x=-8-280$$

Subtract 280 from -8.

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

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.

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