Solve xx+60x+619=0

$$xx+60x+619=0$$

Solve for x (complex solution)

$$x=\sqrt{281}-30\approx -13.236945386$$

$$x=-\left(\sqrt{281}+30\right)\approx -46.763054614$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x^{2}+60x+619=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 60 for b, and 619 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 619}}{2}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\times 619}}{2}$$

Multiply -4 times 619.

$$x=\frac{-60±\sqrt{3600-2476}}{2}$$

Add 3600 to -2476.

$$x=\frac{-60±\sqrt{1124}}{2}$$

Take the square root of 1124.

$$x=\frac{-60±2\sqrt{281}}{2}$$

Now solve the equation x=\frac{-60±2\sqrt{281}}{2} when ± is plus. Add -60 to 2\sqrt{281}.

$$x=\frac{2\sqrt{281}-60}{2}$$

Divide -60+2\sqrt{281} by 2.

$$x=\sqrt{281}-30$$

Now solve the equation x=\frac{-60±2\sqrt{281}}{2} when ± is minus. Subtract 2\sqrt{281} from -60.

$$x=\frac{-2\sqrt{281}-60}{2}$$

Divide -60-2\sqrt{281} by 2.

$$x=-\sqrt{281}-30$$

The equation is now solved.

$$x=\sqrt{281}-30 x=-\sqrt{281}-30$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x^{2}+60x+619=0$$

Subtract 619 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+60x=-619$$

Divide 60, the coefficient of the x term, by 2 to get 30. Then add the square of 30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+60x+30^{2}=-619+30^{2}$$

Square 30.

$$x^{2}+60x+900=-619+900$$

Add -619 to 900.

$$x^{2}+60x+900=281$$

Factor x^{2}+60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+30\right)^{2}=281$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{281}$$

Simplify.

$$x+30=\sqrt{281} x+30=-\sqrt{281}$$

Subtract 30 from both sides of the equation.

$$x=\sqrt{281}-30 x=-\sqrt{281}-30$$

Solve for x

$$x=\sqrt{281}-30\approx -13.236945386$$

$$x=-\sqrt{281}-30\approx -46.763054614$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x^{2}+60x+619=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 60 for b, and 619 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 619}}{2}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\times 619}}{2}$$

Multiply -4 times 619.

$$x=\frac{-60±\sqrt{3600-2476}}{2}$$

Add 3600 to -2476.

$$x=\frac{-60±\sqrt{1124}}{2}$$

Take the square root of 1124.

$$x=\frac{-60±2\sqrt{281}}{2}$$

Now solve the equation x=\frac{-60±2\sqrt{281}}{2} when ± is plus. Add -60 to 2\sqrt{281}.

$$x=\frac{2\sqrt{281}-60}{2}$$

Divide -60+2\sqrt{281} by 2.

$$x=\sqrt{281}-30$$

Now solve the equation x=\frac{-60±2\sqrt{281}}{2} when ± is minus. Subtract 2\sqrt{281} from -60.

$$x=\frac{-2\sqrt{281}-60}{2}$$

Divide -60-2\sqrt{281} by 2.

$$x=-\sqrt{281}-30$$

The equation is now solved.

$$x=\sqrt{281}-30 x=-\sqrt{281}-30$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x^{2}+60x+619=0$$

Subtract 619 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+60x=-619$$

Divide 60, the coefficient of the x term, by 2 to get 30. Then add the square of 30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+60x+30^{2}=-619+30^{2}$$

Square 30.

$$x^{2}+60x+900=-619+900$$

Add -619 to 900.

$$x^{2}+60x+900=281$$

Factor x^{2}+60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+30\right)^{2}=281$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{281}$$

Simplify.

$$x+30=\sqrt{281} x+30=-\sqrt{281}$$

Subtract 30 from both sides of the equation.

$$x=\sqrt{281}-30 x=-\sqrt{281}-30$$