Solve x^2+10x+5=-7x^2-14x-10

$$x ^ { 2 } + 10 x + 5 = - 7 x ^ { 2 } - 14 x - 10$$

Solve for x

$$x=\frac{\sqrt{6}}{4}-\frac{3}{2}\approx -0.887627564$$

$$x=-\frac{\sqrt{6}}{4}-\frac{3}{2}\approx -2.112372436$$

Steps Using the Quadratic Formula

Add 7x^{2} to both sides.

$$x^{2}+10x+5+7x^{2}=-14x-10$$

Combine x^{2} and 7x^{2} to get 8x^{2}.

$$8x^{2}+10x+5=-14x-10$$

Add 14x to both sides.

$$8x^{2}+10x+5+14x=-10$$

Combine 10x and 14x to get 24x.

$$8x^{2}+24x+5=-10$$

Add 10 to both sides.

$$8x^{2}+24x+5+10=0$$

Add 5 and 10 to get 15.

$$8x^{2}+24x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 24 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 8\times 15}}{2\times 8}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 8\times 15}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-24±\sqrt{576-32\times 15}}{2\times 8}$$

Multiply -32 times 15.

$$x=\frac{-24±\sqrt{576-480}}{2\times 8}$$

Add 576 to -480.

$$x=\frac{-24±\sqrt{96}}{2\times 8}$$

Take the square root of 96.

$$x=\frac{-24±4\sqrt{6}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-24±4\sqrt{6}}{16}$$

Now solve the equation x=\frac{-24±4\sqrt{6}}{16} when ± is plus. Add -24 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}-24}{16}$$

Divide -24+4\sqrt{6} by 16.

$$x=\frac{\sqrt{6}}{4}-\frac{3}{2}$$

Now solve the equation x=\frac{-24±4\sqrt{6}}{16} when ± is minus. Subtract 4\sqrt{6} from -24.

$$x=\frac{-4\sqrt{6}-24}{16}$$

Divide -24-4\sqrt{6} by 16.

$$x=-\frac{\sqrt{6}}{4}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{6}}{4}-\frac{3}{2} x=-\frac{\sqrt{6}}{4}-\frac{3}{2}$$

Steps for Completing the Square

Add 7x^{2} to both sides.

$$x^{2}+10x+5+7x^{2}=-14x-10$$

Combine x^{2} and 7x^{2} to get 8x^{2}.

$$8x^{2}+10x+5=-14x-10$$

Add 14x to both sides.

$$8x^{2}+10x+5+14x=-10$$

Combine 10x and 14x to get 24x.

$$8x^{2}+24x+5=-10$$

Subtract 5 from both sides.

$$8x^{2}+24x=-10-5$$

Subtract 5 from -10 to get -15.

$$8x^{2}+24x=-15$$

Divide both sides by 8.

$$\frac{8x^{2}+24x}{8}=-\frac{15}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{24}{8}x=-\frac{15}{8}$$

Divide 24 by 8.

$$x^{2}+3x=-\frac{15}{8}$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-\frac{15}{8}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-\frac{15}{8}+\frac{9}{4}$$

Add -\frac{15}{8} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{3}{8}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{3}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{3}{8}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{6}}{4} x+\frac{3}{2}=-\frac{\sqrt{6}}{4}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{6}}{4}-\frac{3}{2} x=-\frac{\sqrt{6}}{4}-\frac{3}{2}$$