Solve x^2+76x-76*112.6=0

$${ x }^{ 2 } +76x-76 \times 112.6=0$$

Solve for x

$$x=\frac{2\sqrt{62510}}{5}-38\approx 62.00799968$$

$$x=-\frac{2\sqrt{62510}}{5}-38\approx -138.00799968$$

Steps Using the Quadratic Formula

Multiply 76 and 112.6 to get 8557.6.

$$x^{2}+76x-8557.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 76 for b, and -8557.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-8557.6\right)}}{2}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-8557.6\right)}}{2}$$

Multiply -4 times -8557.6.

$$x=\frac{-76±\sqrt{5776+34230.4}}{2}$$

Add 5776 to 34230.4.

$$x=\frac{-76±\sqrt{40006.4}}{2}$$

Take the square root of 40006.4.

$$x=\frac{-76±\frac{4\sqrt{62510}}{5}}{2}$$

Now solve the equation x=\frac{-76±\frac{4\sqrt{62510}}{5}}{2} when ± is plus. Add -76 to \frac{4\sqrt{62510}}{5}.

$$x=\frac{\frac{4\sqrt{62510}}{5}-76}{2}$$

Divide -76+\frac{4\sqrt{62510}}{5} by 2.

$$x=\frac{2\sqrt{62510}}{5}-38$$

Now solve the equation x=\frac{-76±\frac{4\sqrt{62510}}{5}}{2} when ± is minus. Subtract \frac{4\sqrt{62510}}{5} from -76.

$$x=\frac{-\frac{4\sqrt{62510}}{5}-76}{2}$$

Divide -76-\frac{4\sqrt{62510}}{5} by 2.

$$x=-\frac{2\sqrt{62510}}{5}-38$$

The equation is now solved.

$$x=\frac{2\sqrt{62510}}{5}-38 x=-\frac{2\sqrt{62510}}{5}-38$$

Steps for Completing the Square

Multiply 76 and 112.6 to get 8557.6.

$$x^{2}+76x-8557.6=0$$

Add 8557.6 to both sides. Anything plus zero gives itself.

$$x^{2}+76x=8557.6$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=8557.6+38^{2}$$

Square 38.

$$x^{2}+76x+1444=8557.6+1444$$

Add 8557.6 to 1444.

$$x^{2}+76x+1444=10001.6$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=10001.6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{10001.6}$$

Simplify.

$$x+38=\frac{2\sqrt{62510}}{5} x+38=-\frac{2\sqrt{62510}}{5}$$

Subtract 38 from both sides of the equation.

$$x=\frac{2\sqrt{62510}}{5}-38 x=-\frac{2\sqrt{62510}}{5}-38$$