Solve x^2+32x=42

$$x ^ { 2 } + 32 x = 42$$

Solve for x (complex solution)

$$x=\sqrt{298}-16\approx 1.262676502$$

$$x=-\left(\sqrt{298}+16\right)\approx -33.262676502$$

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.

$$x^{2}+32x=42$$

Subtract 42 from both sides of the equation.

$$x^{2}+32x-42=42-42$$

Subtracting 42 from itself leaves 0.

$$x^{2}+32x-42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 32 for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-42\right)}}{2}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-42\right)}}{2}$$

Multiply -4 times -42.

$$x=\frac{-32±\sqrt{1024+168}}{2}$$

Add 1024 to 168.

$$x=\frac{-32±\sqrt{1192}}{2}$$

Take the square root of 1192.

$$x=\frac{-32±2\sqrt{298}}{2}$$

Now solve the equation x=\frac{-32±2\sqrt{298}}{2} when ± is plus. Add -32 to 2\sqrt{298}.

$$x=\frac{2\sqrt{298}-32}{2}$$

Divide -32+2\sqrt{298} by 2.

$$x=\sqrt{298}-16$$

Now solve the equation x=\frac{-32±2\sqrt{298}}{2} when ± is minus. Subtract 2\sqrt{298} from -32.

$$x=\frac{-2\sqrt{298}-32}{2}$$

Divide -32-2\sqrt{298} by 2.

$$x=-\sqrt{298}-16$$

The equation is now solved.

$$x=\sqrt{298}-16 x=-\sqrt{298}-16$$

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.

$$x^{2}+32x=42$$

Divide 32, the coefficient of the x term, by 2 to get 16. Then add the square of 16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+32x+16^{2}=42+16^{2}$$

Square 16.

$$x^{2}+32x+256=42+256$$

Add 42 to 256.

$$x^{2}+32x+256=298$$

Factor x^{2}+32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+16\right)^{2}=298$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+16\right)^{2}}=\sqrt{298}$$

Simplify.

$$x+16=\sqrt{298} x+16=-\sqrt{298}$$

Subtract 16 from both sides of the equation.

$$x=\sqrt{298}-16 x=-\sqrt{298}-16$$

Solve for x

$$x=\sqrt{298}-16\approx 1.262676502$$

$$x=-\sqrt{298}-16\approx -33.262676502$$

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.

$$x^{2}+32x=42$$

Subtract 42 from both sides of the equation.

$$x^{2}+32x-42=42-42$$

Subtracting 42 from itself leaves 0.

$$x^{2}+32x-42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 32 for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-42\right)}}{2}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-42\right)}}{2}$$

Multiply -4 times -42.

$$x=\frac{-32±\sqrt{1024+168}}{2}$$

Add 1024 to 168.

$$x=\frac{-32±\sqrt{1192}}{2}$$

Take the square root of 1192.

$$x=\frac{-32±2\sqrt{298}}{2}$$

Now solve the equation x=\frac{-32±2\sqrt{298}}{2} when ± is plus. Add -32 to 2\sqrt{298}.

$$x=\frac{2\sqrt{298}-32}{2}$$

Divide -32+2\sqrt{298} by 2.

$$x=\sqrt{298}-16$$

Now solve the equation x=\frac{-32±2\sqrt{298}}{2} when ± is minus. Subtract 2\sqrt{298} from -32.

$$x=\frac{-2\sqrt{298}-32}{2}$$

Divide -32-2\sqrt{298} by 2.

$$x=-\sqrt{298}-16$$

The equation is now solved.

$$x=\sqrt{298}-16 x=-\sqrt{298}-16$$

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.

$$x^{2}+32x=42$$

Divide 32, the coefficient of the x term, by 2 to get 16. Then add the square of 16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+32x+16^{2}=42+16^{2}$$

Square 16.

$$x^{2}+32x+256=42+256$$

Add 42 to 256.

$$x^{2}+32x+256=298$$

Factor x^{2}+32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+16\right)^{2}=298$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+16\right)^{2}}=\sqrt{298}$$

Simplify.

$$x+16=\sqrt{298} x+16=-\sqrt{298}$$

Subtract 16 from both sides of the equation.

$$x=\sqrt{298}-16 x=-\sqrt{298}-16$$