Solve x^2+488x+89=0

$${ x }^{ 2 } +488x+89=0$$

Solve for x (complex solution)

$$x=\sqrt{59447}-244\approx -0.182445259$$

$$x=-\left(\sqrt{59447}+244\right)\approx -487.817554741$$

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.

$$x^{2}+488x+89=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 488 for b, and 89 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-488±\sqrt{488^{2}-4\times 89}}{2}$$

Square 488.

$$x=\frac{-488±\sqrt{238144-4\times 89}}{2}$$

Multiply -4 times 89.

$$x=\frac{-488±\sqrt{238144-356}}{2}$$

Add 238144 to -356.

$$x=\frac{-488±\sqrt{237788}}{2}$$

Take the square root of 237788.

$$x=\frac{-488±2\sqrt{59447}}{2}$$

Now solve the equation x=\frac{-488±2\sqrt{59447}}{2} when ± is plus. Add -488 to 2\sqrt{59447}.

$$x=\frac{2\sqrt{59447}-488}{2}$$

Divide -488+2\sqrt{59447} by 2.

$$x=\sqrt{59447}-244$$

Now solve the equation x=\frac{-488±2\sqrt{59447}}{2} when ± is minus. Subtract 2\sqrt{59447} from -488.

$$x=\frac{-2\sqrt{59447}-488}{2}$$

Divide -488-2\sqrt{59447} by 2.

$$x=-\sqrt{59447}-244$$

The equation is now solved.

$$x=\sqrt{59447}-244 x=-\sqrt{59447}-244$$

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.

$$x^{2}+488x+89=0$$

Subtract 89 from both sides of the equation.

$$x^{2}+488x+89-89=-89$$

Subtracting 89 from itself leaves 0.

$$x^{2}+488x=-89$$

Divide 488, the coefficient of the x term, by 2 to get 244. Then add the square of 244 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+488x+244^{2}=-89+244^{2}$$

Square 244.

$$x^{2}+488x+59536=-89+59536$$

Add -89 to 59536.

$$x^{2}+488x+59536=59447$$

Factor x^{2}+488x+59536. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+244\right)^{2}=59447$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+244\right)^{2}}=\sqrt{59447}$$

Simplify.

$$x+244=\sqrt{59447} x+244=-\sqrt{59447}$$

Subtract 244 from both sides of the equation.

$$x=\sqrt{59447}-244 x=-\sqrt{59447}-244$$

Solve for x

$$x=\sqrt{59447}-244\approx -0.182445259$$

$$x=-\sqrt{59447}-244\approx -487.817554741$$

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.

$$x^{2}+488x+89=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 488 for b, and 89 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-488±\sqrt{488^{2}-4\times 89}}{2}$$

Square 488.

$$x=\frac{-488±\sqrt{238144-4\times 89}}{2}$$

Multiply -4 times 89.

$$x=\frac{-488±\sqrt{238144-356}}{2}$$

Add 238144 to -356.

$$x=\frac{-488±\sqrt{237788}}{2}$$

Take the square root of 237788.

$$x=\frac{-488±2\sqrt{59447}}{2}$$

Now solve the equation x=\frac{-488±2\sqrt{59447}}{2} when ± is plus. Add -488 to 2\sqrt{59447}.

$$x=\frac{2\sqrt{59447}-488}{2}$$

Divide -488+2\sqrt{59447} by 2.

$$x=\sqrt{59447}-244$$

Now solve the equation x=\frac{-488±2\sqrt{59447}}{2} when ± is minus. Subtract 2\sqrt{59447} from -488.

$$x=\frac{-2\sqrt{59447}-488}{2}$$

Divide -488-2\sqrt{59447} by 2.

$$x=-\sqrt{59447}-244$$

The equation is now solved.

$$x=\sqrt{59447}-244 x=-\sqrt{59447}-244$$

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.

$$x^{2}+488x+89=0$$

Subtract 89 from both sides of the equation.

$$x^{2}+488x+89-89=-89$$

Subtracting 89 from itself leaves 0.

$$x^{2}+488x=-89$$

Divide 488, the coefficient of the x term, by 2 to get 244. Then add the square of 244 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+488x+244^{2}=-89+244^{2}$$

Square 244.

$$x^{2}+488x+59536=-89+59536$$

Add -89 to 59536.

$$x^{2}+488x+59536=59447$$

Factor x^{2}+488x+59536. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+244\right)^{2}=59447$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+244\right)^{2}}=\sqrt{59447}$$

Simplify.

$$x+244=\sqrt{59447} x+244=-\sqrt{59447}$$

Subtract 244 from both sides of the equation.

$$x=\sqrt{59447}-244 x=-\sqrt{59447}-244$$