Solve x^2-503.95x-127356.052=0

$${ x }^{ 2 } -503.95x-127356.052=0$$

Solve for x

$$x=\frac{\sqrt{7633898105}}{200}+251.975\approx 688.835907641$$

$$x=-\frac{\sqrt{7633898105}}{200}+251.975\approx -184.885907641$$

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.

$$x^{2}-503.95x-127356.052=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -503.95 for b, and -127356.052 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-503.95\right)±\sqrt{\left(-503.95\right)^{2}-4\left(-127356.052\right)}}{2}$$

Square -503.95 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-503.95\right)±\sqrt{253965.6025-4\left(-127356.052\right)}}{2}$$

Multiply -4 times -127356.052.

$$x=\frac{-\left(-503.95\right)±\sqrt{253965.6025+509424.208}}{2}$$

Add 253965.6025 to 509424.208 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-503.95\right)±\sqrt{763389.8105}}{2}$$

Take the square root of 763389.8105.

$$x=\frac{-\left(-503.95\right)±\frac{\sqrt{7633898105}}{100}}{2}$$

The opposite of -503.95 is 503.95.

$$x=\frac{503.95±\frac{\sqrt{7633898105}}{100}}{2}$$

Now solve the equation x=\frac{503.95±\frac{\sqrt{7633898105}}{100}}{2} when ± is plus. Add 503.95 to \frac{\sqrt{7633898105}}{100}.

$$x=\frac{\frac{\sqrt{7633898105}}{100}+\frac{10079}{20}}{2}$$

Divide \frac{10079}{20}+\frac{\sqrt{7633898105}}{100} by 2.

$$x=\frac{\sqrt{7633898105}}{200}+\frac{10079}{40}$$

Now solve the equation x=\frac{503.95±\frac{\sqrt{7633898105}}{100}}{2} when ± is minus. Subtract \frac{\sqrt{7633898105}}{100} from 503.95.

$$x=\frac{-\frac{\sqrt{7633898105}}{100}+\frac{10079}{20}}{2}$$

Divide \frac{10079}{20}-\frac{\sqrt{7633898105}}{100} by 2.

$$x=-\frac{\sqrt{7633898105}}{200}+\frac{10079}{40}$$

The equation is now solved.

$$x=\frac{\sqrt{7633898105}}{200}+\frac{10079}{40} x=-\frac{\sqrt{7633898105}}{200}+\frac{10079}{40}$$

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.

$$x^{2}-503.95x-127356.052=0$$

Add 127356.052 to both sides of the equation.

$$x^{2}-503.95x-127356.052-\left(-127356.052\right)=-\left(-127356.052\right)$$

Subtracting -127356.052 from itself leaves 0.

$$x^{2}-503.95x=-\left(-127356.052\right)$$

Subtract -127356.052 from 0.

$$x^{2}-503.95x=127356.052$$

Divide -503.95, the coefficient of the x term, by 2 to get -251.975. Then add the square of -251.975 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-503.95x+\left(-251.975\right)^{2}=127356.052+\left(-251.975\right)^{2}$$

Square -251.975 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-503.95x+63491.400625=127356.052+63491.400625$$

Add 127356.052 to 63491.400625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-503.95x+63491.400625=190847.452625$$

Factor x^{2}-503.95x+63491.400625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-251.975\right)^{2}=190847.452625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-251.975\right)^{2}}=\sqrt{190847.452625}$$

Simplify.

$$x-251.975=\frac{\sqrt{7633898105}}{200} x-251.975=-\frac{\sqrt{7633898105}}{200}$$

Add 251.975 to both sides of the equation.

$$x=\frac{\sqrt{7633898105}}{200}+\frac{10079}{40} x=-\frac{\sqrt{7633898105}}{200}+\frac{10079}{40}$$