Solve x^2+20x=45

$$x ^ { 2 } + 20 x = 45$$

Solve for x (complex solution)

$$x=\sqrt{145}-10\approx 2.041594579$$

$$x=-\left(\sqrt{145}+10\right)\approx -22.041594579$$

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.

$$x^{2}+20x=45$$

Subtract 45 from both sides of the equation.

$$x^{2}+20x-45=45-45$$

Subtracting 45 from itself leaves 0.

$$x^{2}+20x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-45\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-45\right)}}{2}$$

Multiply -4 times -45.

$$x=\frac{-20±\sqrt{400+180}}{2}$$

Add 400 to 180.

$$x=\frac{-20±\sqrt{580}}{2}$$

Take the square root of 580.

$$x=\frac{-20±2\sqrt{145}}{2}$$

Now solve the equation x=\frac{-20±2\sqrt{145}}{2} when ± is plus. Add -20 to 2\sqrt{145}.

$$x=\frac{2\sqrt{145}-20}{2}$$

Divide -20+2\sqrt{145} by 2.

$$x=\sqrt{145}-10$$

Now solve the equation x=\frac{-20±2\sqrt{145}}{2} when ± is minus. Subtract 2\sqrt{145} from -20.

$$x=\frac{-2\sqrt{145}-20}{2}$$

Divide -20-2\sqrt{145} by 2.

$$x=-\sqrt{145}-10$$

The equation is now solved.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

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.

$$x^{2}+20x=45$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=45+10^{2}$$

Square 10.

$$x^{2}+20x+100=45+100$$

Add 45 to 100.

$$x^{2}+20x+100=145$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=145$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{145}$$

Simplify.

$$x+10=\sqrt{145} x+10=-\sqrt{145}$$

Subtract 10 from both sides of the equation.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

Solve for x

$$x=\sqrt{145}-10\approx 2.041594579$$

$$x=-\sqrt{145}-10\approx -22.041594579$$

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.

$$x^{2}+20x=45$$

Subtract 45 from both sides of the equation.

$$x^{2}+20x-45=45-45$$

Subtracting 45 from itself leaves 0.

$$x^{2}+20x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-45\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-45\right)}}{2}$$

Multiply -4 times -45.

$$x=\frac{-20±\sqrt{400+180}}{2}$$

Add 400 to 180.

$$x=\frac{-20±\sqrt{580}}{2}$$

Take the square root of 580.

$$x=\frac{-20±2\sqrt{145}}{2}$$

Now solve the equation x=\frac{-20±2\sqrt{145}}{2} when ± is plus. Add -20 to 2\sqrt{145}.

$$x=\frac{2\sqrt{145}-20}{2}$$

Divide -20+2\sqrt{145} by 2.

$$x=\sqrt{145}-10$$

Now solve the equation x=\frac{-20±2\sqrt{145}}{2} when ± is minus. Subtract 2\sqrt{145} from -20.

$$x=\frac{-2\sqrt{145}-20}{2}$$

Divide -20-2\sqrt{145} by 2.

$$x=-\sqrt{145}-10$$

The equation is now solved.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

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.

$$x^{2}+20x=45$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=45+10^{2}$$

Square 10.

$$x^{2}+20x+100=45+100$$

Add 45 to 100.

$$x^{2}+20x+100=145$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=145$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{145}$$

Simplify.

$$x+10=\sqrt{145} x+10=-\sqrt{145}$$

Subtract 10 from both sides of the equation.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$