Solve x^2+34x=285

$$x ^ { 2 } + 34 x = 285$$

Solve for x (complex solution)

$$x=\sqrt{574}-17\approx 6.958297101$$

$$x=-\left(\sqrt{574}+17\right)\approx -40.958297101$$

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.

$$x^{2}+34x=285$$

Subtract 285 from both sides of the equation.

$$x^{2}+34x-285=285-285$$

Subtracting 285 from itself leaves 0.

$$x^{2}+34x-285=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 34 for b, and -285 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-285\right)}}{2}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-285\right)}}{2}$$

Multiply -4 times -285.

$$x=\frac{-34±\sqrt{1156+1140}}{2}$$

Add 1156 to 1140.

$$x=\frac{-34±\sqrt{2296}}{2}$$

Take the square root of 2296.

$$x=\frac{-34±2\sqrt{574}}{2}$$

Now solve the equation x=\frac{-34±2\sqrt{574}}{2} when ± is plus. Add -34 to 2\sqrt{574}.

$$x=\frac{2\sqrt{574}-34}{2}$$

Divide -34+2\sqrt{574} by 2.

$$x=\sqrt{574}-17$$

Now solve the equation x=\frac{-34±2\sqrt{574}}{2} when ± is minus. Subtract 2\sqrt{574} from -34.

$$x=\frac{-2\sqrt{574}-34}{2}$$

Divide -34-2\sqrt{574} by 2.

$$x=-\sqrt{574}-17$$

The equation is now solved.

$$x=\sqrt{574}-17 x=-\sqrt{574}-17$$

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.

$$x^{2}+34x=285$$

Divide 34, the coefficient of the x term, by 2 to get 17. Then add the square of 17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+34x+17^{2}=285+17^{2}$$

Square 17.

$$x^{2}+34x+289=285+289$$

Add 285 to 289.

$$x^{2}+34x+289=574$$

Factor x^{2}+34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+17\right)^{2}=574$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{574}$$

Simplify.

$$x+17=\sqrt{574} x+17=-\sqrt{574}$$

Subtract 17 from both sides of the equation.

$$x=\sqrt{574}-17 x=-\sqrt{574}-17$$

Solve for x

$$x=\sqrt{574}-17\approx 6.958297101$$

$$x=-\sqrt{574}-17\approx -40.958297101$$

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.

$$x^{2}+34x=285$$

Subtract 285 from both sides of the equation.

$$x^{2}+34x-285=285-285$$

Subtracting 285 from itself leaves 0.

$$x^{2}+34x-285=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 34 for b, and -285 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-285\right)}}{2}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-285\right)}}{2}$$

Multiply -4 times -285.

$$x=\frac{-34±\sqrt{1156+1140}}{2}$$

Add 1156 to 1140.

$$x=\frac{-34±\sqrt{2296}}{2}$$

Take the square root of 2296.

$$x=\frac{-34±2\sqrt{574}}{2}$$

Now solve the equation x=\frac{-34±2\sqrt{574}}{2} when ± is plus. Add -34 to 2\sqrt{574}.

$$x=\frac{2\sqrt{574}-34}{2}$$

Divide -34+2\sqrt{574} by 2.

$$x=\sqrt{574}-17$$

Now solve the equation x=\frac{-34±2\sqrt{574}}{2} when ± is minus. Subtract 2\sqrt{574} from -34.

$$x=\frac{-2\sqrt{574}-34}{2}$$

Divide -34-2\sqrt{574} by 2.

$$x=-\sqrt{574}-17$$

The equation is now solved.

$$x=\sqrt{574}-17 x=-\sqrt{574}-17$$

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.

$$x^{2}+34x=285$$

Divide 34, the coefficient of the x term, by 2 to get 17. Then add the square of 17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+34x+17^{2}=285+17^{2}$$

Square 17.

$$x^{2}+34x+289=285+289$$

Add 285 to 289.

$$x^{2}+34x+289=574$$

Factor x^{2}+34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+17\right)^{2}=574$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{574}$$

Simplify.

$$x+17=\sqrt{574} x+17=-\sqrt{574}$$

Subtract 17 from both sides of the equation.

$$x=\sqrt{574}-17 x=-\sqrt{574}-17$$