Solve x^2+570x-6610=0

$${ x }^{ 2 } +570x-6610=0$$

Solve for x (complex solution)

$$x=\sqrt{87835}-285\approx 11.369701555$$

$$x=-\left(\sqrt{87835}+285\right)\approx -581.369701555$$

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.

$$x^{2}+570x-6610=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 570 for b, and -6610 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-570±\sqrt{570^{2}-4\left(-6610\right)}}{2}$$

Square 570.

$$x=\frac{-570±\sqrt{324900-4\left(-6610\right)}}{2}$$

Multiply -4 times -6610.

$$x=\frac{-570±\sqrt{324900+26440}}{2}$$

Add 324900 to 26440.

$$x=\frac{-570±\sqrt{351340}}{2}$$

Take the square root of 351340.

$$x=\frac{-570±2\sqrt{87835}}{2}$$

Now solve the equation x=\frac{-570±2\sqrt{87835}}{2} when ± is plus. Add -570 to 2\sqrt{87835}.

$$x=\frac{2\sqrt{87835}-570}{2}$$

Divide -570+2\sqrt{87835} by 2.

$$x=\sqrt{87835}-285$$

Now solve the equation x=\frac{-570±2\sqrt{87835}}{2} when ± is minus. Subtract 2\sqrt{87835} from -570.

$$x=\frac{-2\sqrt{87835}-570}{2}$$

Divide -570-2\sqrt{87835} by 2.

$$x=-\sqrt{87835}-285$$

The equation is now solved.

$$x=\sqrt{87835}-285 x=-\sqrt{87835}-285$$

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.

$$x^{2}+570x-6610=0$$

Add 6610 to both sides of the equation.

$$x^{2}+570x-6610-\left(-6610\right)=-\left(-6610\right)$$

Subtracting -6610 from itself leaves 0.

$$x^{2}+570x=-\left(-6610\right)$$

Subtract -6610 from 0.

$$x^{2}+570x=6610$$

Divide 570, the coefficient of the x term, by 2 to get 285. Then add the square of 285 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+570x+285^{2}=6610+285^{2}$$

Square 285.

$$x^{2}+570x+81225=6610+81225$$

Add 6610 to 81225.

$$x^{2}+570x+81225=87835$$

Factor x^{2}+570x+81225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+285\right)^{2}=87835$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+285\right)^{2}}=\sqrt{87835}$$

Simplify.

$$x+285=\sqrt{87835} x+285=-\sqrt{87835}$$

Subtract 285 from both sides of the equation.

$$x=\sqrt{87835}-285 x=-\sqrt{87835}-285$$

Solve for x

$$x=\sqrt{87835}-285\approx 11.369701555$$

$$x=-\sqrt{87835}-285\approx -581.369701555$$

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.

$$x^{2}+570x-6610=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 570 for b, and -6610 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-570±\sqrt{570^{2}-4\left(-6610\right)}}{2}$$

Square 570.

$$x=\frac{-570±\sqrt{324900-4\left(-6610\right)}}{2}$$

Multiply -4 times -6610.

$$x=\frac{-570±\sqrt{324900+26440}}{2}$$

Add 324900 to 26440.

$$x=\frac{-570±\sqrt{351340}}{2}$$

Take the square root of 351340.

$$x=\frac{-570±2\sqrt{87835}}{2}$$

Now solve the equation x=\frac{-570±2\sqrt{87835}}{2} when ± is plus. Add -570 to 2\sqrt{87835}.

$$x=\frac{2\sqrt{87835}-570}{2}$$

Divide -570+2\sqrt{87835} by 2.

$$x=\sqrt{87835}-285$$

Now solve the equation x=\frac{-570±2\sqrt{87835}}{2} when ± is minus. Subtract 2\sqrt{87835} from -570.

$$x=\frac{-2\sqrt{87835}-570}{2}$$

Divide -570-2\sqrt{87835} by 2.

$$x=-\sqrt{87835}-285$$

The equation is now solved.

$$x=\sqrt{87835}-285 x=-\sqrt{87835}-285$$

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.

$$x^{2}+570x-6610=0$$

Add 6610 to both sides of the equation.

$$x^{2}+570x-6610-\left(-6610\right)=-\left(-6610\right)$$

Subtracting -6610 from itself leaves 0.

$$x^{2}+570x=-\left(-6610\right)$$

Subtract -6610 from 0.

$$x^{2}+570x=6610$$

Divide 570, the coefficient of the x term, by 2 to get 285. Then add the square of 285 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+570x+285^{2}=6610+285^{2}$$

Square 285.

$$x^{2}+570x+81225=6610+81225$$

Add 6610 to 81225.

$$x^{2}+570x+81225=87835$$

Factor x^{2}+570x+81225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+285\right)^{2}=87835$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+285\right)^{2}}=\sqrt{87835}$$

Simplify.

$$x+285=\sqrt{87835} x+285=-\sqrt{87835}$$

Subtract 285 from both sides of the equation.

$$x=\sqrt{87835}-285 x=-\sqrt{87835}-285$$