Solve 2x^2+157x-2486=0

$$2 { x }^{ 2 } +157x-2486=0$$

Solve for x

$$x = \frac{\sqrt{44537} - 157}{4} \approx 13.509477822$$

$$x=\frac{-\sqrt{44537}-157}{4}\approx -92.009477822$$

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.

$$2x^{2}+157x-2486=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 157 for b, and -2486 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-157±\sqrt{157^{2}-4\times 2\left(-2486\right)}}{2\times 2}$$

Square 157.

$$x=\frac{-157±\sqrt{24649-4\times 2\left(-2486\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-157±\sqrt{24649-8\left(-2486\right)}}{2\times 2}$$

Multiply -8 times -2486.

$$x=\frac{-157±\sqrt{24649+19888}}{2\times 2}$$

Add 24649 to 19888.

$$x=\frac{-157±\sqrt{44537}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-157±\sqrt{44537}}{4}$$

Now solve the equation x=\frac{-157±\sqrt{44537}}{4} when ± is plus. Add -157 to \sqrt{44537}.

$$x=\frac{\sqrt{44537}-157}{4}$$

Now solve the equation x=\frac{-157±\sqrt{44537}}{4} when ± is minus. Subtract \sqrt{44537} from -157.

$$x=\frac{-\sqrt{44537}-157}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{44537}-157}{4} x=\frac{-\sqrt{44537}-157}{4}$$

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.

$$2x^{2}+157x-2486=0$$

Add 2486 to both sides of the equation.

$$2x^{2}+157x-2486-\left(-2486\right)=-\left(-2486\right)$$

Subtracting -2486 from itself leaves 0.

$$2x^{2}+157x=-\left(-2486\right)$$

Subtract -2486 from 0.

$$2x^{2}+157x=2486$$

Divide both sides by 2.

$$\frac{2x^{2}+157x}{2}=\frac{2486}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{157}{2}x=\frac{2486}{2}$$

Divide 2486 by 2.

$$x^{2}+\frac{157}{2}x=1243$$

Divide \frac{157}{2}, the coefficient of the x term, by 2 to get \frac{157}{4}. Then add the square of \frac{157}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{157}{2}x+\left(\frac{157}{4}\right)^{2}=1243+\left(\frac{157}{4}\right)^{2}$$

Square \frac{157}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{157}{2}x+\frac{24649}{16}=1243+\frac{24649}{16}$$

Add 1243 to \frac{24649}{16}.

$$x^{2}+\frac{157}{2}x+\frac{24649}{16}=\frac{44537}{16}$$

Factor x^{2}+\frac{157}{2}x+\frac{24649}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{157}{4}\right)^{2}=\frac{44537}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{157}{4}\right)^{2}}=\sqrt{\frac{44537}{16}}$$

Simplify.

$$x+\frac{157}{4}=\frac{\sqrt{44537}}{4} x+\frac{157}{4}=-\frac{\sqrt{44537}}{4}$$

Subtract \frac{157}{4} from both sides of the equation.

$$x=\frac{\sqrt{44537}-157}{4} x=\frac{-\sqrt{44537}-157}{4}$$