Solve x^2+30x=225-(20)

$$x ^ { 2 } + 30 x = 225 - ( 20 )$$

Solve for x (complex solution)

$$x=\sqrt{430}-15\approx 5.736441353$$

$$x=-\left(\sqrt{430}+15\right)\approx -35.736441353$$

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.

$$x^{2}+30x=205$$

Subtract 205 from both sides of the equation.

$$x^{2}+30x-205=205-205$$

Subtracting 205 from itself leaves 0.

$$x^{2}+30x-205=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30 for b, and -205 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-205\right)}}{2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-205\right)}}{2}$$

Multiply -4 times -205.

$$x=\frac{-30±\sqrt{900+820}}{2}$$

Add 900 to 820.

$$x=\frac{-30±\sqrt{1720}}{2}$$

Take the square root of 1720.

$$x=\frac{-30±2\sqrt{430}}{2}$$

Now solve the equation x=\frac{-30±2\sqrt{430}}{2} when ± is plus. Add -30 to 2\sqrt{430}.

$$x=\frac{2\sqrt{430}-30}{2}$$

Divide -30+2\sqrt{430} by 2.

$$x=\sqrt{430}-15$$

Now solve the equation x=\frac{-30±2\sqrt{430}}{2} when ± is minus. Subtract 2\sqrt{430} from -30.

$$x=\frac{-2\sqrt{430}-30}{2}$$

Divide -30-2\sqrt{430} by 2.

$$x=-\sqrt{430}-15$$

The equation is now solved.

$$x=\sqrt{430}-15 x=-\sqrt{430}-15$$

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.

$$x^{2}+30x=205$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+30x+15^{2}=205+15^{2}$$

Square 15.

$$x^{2}+30x+225=205+225$$

Add 205 to 225.

$$x^{2}+30x+225=430$$

Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15\right)^{2}=430$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{430}$$

Simplify.

$$x+15=\sqrt{430} x+15=-\sqrt{430}$$

Subtract 15 from both sides of the equation.

$$x=\sqrt{430}-15 x=-\sqrt{430}-15$$

Solve for x

$$x=\sqrt{430}-15\approx 5.736441353$$

$$x=-\sqrt{430}-15\approx -35.736441353$$

Steps Using the Quadratic Formula

Subtract 20 from 225 to get 205.

$$x^{2}+30x=205$$

Subtract 205 from both sides.

$$x^{2}+30x-205=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30 for b, and -205 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-205\right)}}{2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-205\right)}}{2}$$

Multiply -4 times -205.

$$x=\frac{-30±\sqrt{900+820}}{2}$$

Add 900 to 820.

$$x=\frac{-30±\sqrt{1720}}{2}$$

Take the square root of 1720.

$$x=\frac{-30±2\sqrt{430}}{2}$$

Now solve the equation x=\frac{-30±2\sqrt{430}}{2} when ± is plus. Add -30 to 2\sqrt{430}.

$$x=\frac{2\sqrt{430}-30}{2}$$

Divide -30+2\sqrt{430} by 2.

$$x=\sqrt{430}-15$$

Now solve the equation x=\frac{-30±2\sqrt{430}}{2} when ± is minus. Subtract 2\sqrt{430} from -30.

$$x=\frac{-2\sqrt{430}-30}{2}$$

Divide -30-2\sqrt{430} by 2.

$$x=-\sqrt{430}-15$$

The equation is now solved.

$$x=\sqrt{430}-15 x=-\sqrt{430}-15$$

Steps for Completing the Square

Subtract 20 from 225 to get 205.

$$x^{2}+30x=205$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+30x+15^{2}=205+15^{2}$$

Square 15.

$$x^{2}+30x+225=205+225$$

Add 205 to 225.

$$x^{2}+30x+225=430$$

Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15\right)^{2}=430$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{430}$$

Simplify.

$$x+15=\sqrt{430} x+15=-\sqrt{430}$$

Subtract 15 from both sides of the equation.

$$x=\sqrt{430}-15 x=-\sqrt{430}-15$$