Solve a:30=30:(a+30)

$$a : 30 = 30 : ( a + 30 )$$

Solve for a

$$a=15\sqrt{5}-15\approx 18.541019662$$

$$a=-15\sqrt{5}-15\approx -48.541019662$$

Steps Using the Quadratic Formula

Variable a cannot be equal to -30 since division by zero is not defined. Multiply both sides of the equation by 30\left(a+30\right), the least common multiple of 30,a+30.

$$\left(a+30\right)a=30\times 30$$

Use the distributive property to multiply a+30 by a.

$$a^{2}+30a=30\times 30$$

Multiply 30 and 30 to get 900.

$$a^{2}+30a=900$$

Subtract 900 from both sides.

$$a^{2}+30a-900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30 for b, and -900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 30.

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

Multiply -4 times -900.

$$a=\frac{-30±\sqrt{900+3600}}{2}$$

Add 900 to 3600.

$$a=\frac{-30±\sqrt{4500}}{2}$$

Take the square root of 4500.

$$a=\frac{-30±30\sqrt{5}}{2}$$

Now solve the equation a=\frac{-30±30\sqrt{5}}{2} when ± is plus. Add -30 to 30\sqrt{5}.

$$a=\frac{30\sqrt{5}-30}{2}$$

Divide -30+30\sqrt{5} by 2.

$$a=15\sqrt{5}-15$$

Now solve the equation a=\frac{-30±30\sqrt{5}}{2} when ± is minus. Subtract 30\sqrt{5} from -30.

$$a=\frac{-30\sqrt{5}-30}{2}$$

Divide -30-30\sqrt{5} by 2.

$$a=-15\sqrt{5}-15$$

The equation is now solved.

$$a=15\sqrt{5}-15 a=-15\sqrt{5}-15$$

Steps for Completing the Square

Variable a cannot be equal to -30 since division by zero is not defined. Multiply both sides of the equation by 30\left(a+30\right), the least common multiple of 30,a+30.

$$\left(a+30\right)a=30\times 30$$

Use the distributive property to multiply a+30 by a.

$$a^{2}+30a=30\times 30$$

Multiply 30 and 30 to get 900.

$$a^{2}+30a=900$$

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.

$$a^{2}+30a+15^{2}=900+15^{2}$$

Square 15.

$$a^{2}+30a+225=900+225$$

Add 900 to 225.

$$a^{2}+30a+225=1125$$

Factor a^{2}+30a+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}.

$$\left(a+15\right)^{2}=1125$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+15\right)^{2}}=\sqrt{1125}$$

Simplify.

$$a+15=15\sqrt{5} a+15=-15\sqrt{5}$$

Subtract 15 from both sides of the equation.

$$a=15\sqrt{5}-15 a=-15\sqrt{5}-15$$