Solve 8x^2+112x+392=1

$$8 x ^ { 2 } + 112 x + 392 = 1$$

Solve for x

$$x=\frac{\sqrt{2}}{4}-7\approx -6.646446609$$

$$x=-\frac{\sqrt{2}}{4}-7\approx -7.353553391$$

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.

$$8x^{2}+112x+392=1$$

Subtract 1 from both sides of the equation.

$$8x^{2}+112x+392-1=1-1$$

Subtracting 1 from itself leaves 0.

$$8x^{2}+112x+392-1=0$$

Subtract 1 from 392.

$$8x^{2}+112x+391=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 112 for b, and 391 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112±\sqrt{112^{2}-4\times 8\times 391}}{2\times 8}$$

Square 112.

$$x=\frac{-112±\sqrt{12544-4\times 8\times 391}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-112±\sqrt{12544-32\times 391}}{2\times 8}$$

Multiply -32 times 391.

$$x=\frac{-112±\sqrt{12544-12512}}{2\times 8}$$

Add 12544 to -12512.

$$x=\frac{-112±\sqrt{32}}{2\times 8}$$

Take the square root of 32.

$$x=\frac{-112±4\sqrt{2}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-112±4\sqrt{2}}{16}$$

Now solve the equation x=\frac{-112±4\sqrt{2}}{16} when ± is plus. Add -112 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}-112}{16}$$

Divide -112+4\sqrt{2} by 16.

$$x=\frac{\sqrt{2}}{4}-7$$

Now solve the equation x=\frac{-112±4\sqrt{2}}{16} when ± is minus. Subtract 4\sqrt{2} from -112.

$$x=\frac{-4\sqrt{2}-112}{16}$$

Divide -112-4\sqrt{2} by 16.

$$x=-\frac{\sqrt{2}}{4}-7$$

The equation is now solved.

$$x=\frac{\sqrt{2}}{4}-7 x=-\frac{\sqrt{2}}{4}-7$$

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.

$$8x^{2}+112x+392=1$$

Subtract 392 from both sides of the equation.

$$8x^{2}+112x+392-392=1-392$$

Subtracting 392 from itself leaves 0.

$$8x^{2}+112x=1-392$$

Subtract 392 from 1.

$$8x^{2}+112x=-391$$

Divide both sides by 8.

$$\frac{8x^{2}+112x}{8}=-\frac{391}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{112}{8}x=-\frac{391}{8}$$

Divide 112 by 8.

$$x^{2}+14x=-\frac{391}{8}$$

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.

$$x^{2}+14x+7^{2}=-\frac{391}{8}+7^{2}$$

Square 7.

$$x^{2}+14x+49=-\frac{391}{8}+49$$

Add -\frac{391}{8} to 49.

$$x^{2}+14x+49=\frac{1}{8}$$

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}.

$$\left(x+7\right)^{2}=\frac{1}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{\frac{1}{8}}$$

Simplify.

$$x+7=\frac{\sqrt{2}}{4} x+7=-\frac{\sqrt{2}}{4}$$

Subtract 7 from both sides of the equation.

$$x=\frac{\sqrt{2}}{4}-7 x=-\frac{\sqrt{2}}{4}-7$$