Solve 5a/10+a=9a/30+a

$$\frac { 5 a } { 10 + a } = \frac { 9 a } { 30 + a }$$

Solve for a

$$a=15$$

$$a=0$$

Steps Using Factoring

Variable a cannot be equal to any of the values -30,-10 since division by zero is not defined. Multiply both sides of the equation by \left(a+10\right)\left(a+30\right), the least common multiple of 10+a,30+a.

$$\left(a+30\right)\times 5a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply a+30 by 5.

$$\left(5a+150\right)a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply 5a+150 by a.

$$5a^{2}+150a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply a+10 by 9.

$$5a^{2}+150a=\left(9a+90\right)a$$

Use the distributive property to multiply 9a+90 by a.

$$5a^{2}+150a=9a^{2}+90a$$

Subtract 9a^{2} from both sides.

$$5a^{2}+150a-9a^{2}=90a$$

Combine 5a^{2} and -9a^{2} to get -4a^{2}.

$$-4a^{2}+150a=90a$$

Subtract 90a from both sides.

$$-4a^{2}+150a-90a=0$$

Combine 150a and -90a to get 60a.

$$-4a^{2}+60a=0$$

Factor out a.

$$a\left(-4a+60\right)=0$$

To find equation solutions, solve a=0 and -4a+60=0.

$$a=0 a=15$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -30,-10 since division by zero is not defined. Multiply both sides of the equation by \left(a+10\right)\left(a+30\right), the least common multiple of 10+a,30+a.

$$\left(a+30\right)\times 5a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply a+30 by 5.

$$\left(5a+150\right)a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply 5a+150 by a.

$$5a^{2}+150a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply a+10 by 9.

$$5a^{2}+150a=\left(9a+90\right)a$$

Use the distributive property to multiply 9a+90 by a.

$$5a^{2}+150a=9a^{2}+90a$$

Subtract 9a^{2} from both sides.

$$5a^{2}+150a-9a^{2}=90a$$

Combine 5a^{2} and -9a^{2} to get -4a^{2}.

$$-4a^{2}+150a=90a$$

Subtract 90a from both sides.

$$-4a^{2}+150a-90a=0$$

Combine 150a and -90a to get 60a.

$$-4a^{2}+60a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 60 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-60±\sqrt{60^{2}}}{2\left(-4\right)}$$

Take the square root of 60^{2}.

$$a=\frac{-60±60}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{-60±60}{-8}$$

Now solve the equation a=\frac{-60±60}{-8} when ± is plus. Add -60 to 60.

$$a=\frac{0}{-8}$$

Divide 0 by -8.

$$a=0$$

Now solve the equation a=\frac{-60±60}{-8} when ± is minus. Subtract 60 from -60.

$$a=-\frac{120}{-8}$$

Divide -120 by -8.

$$a=15$$

The equation is now solved.

$$a=0 a=15$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -30,-10 since division by zero is not defined. Multiply both sides of the equation by \left(a+10\right)\left(a+30\right), the least common multiple of 10+a,30+a.

$$\left(a+30\right)\times 5a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply a+30 by 5.

$$\left(5a+150\right)a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply 5a+150 by a.

$$5a^{2}+150a=\left(a+10\right)\times 9a$$

Use the distributive property to multiply a+10 by 9.

$$5a^{2}+150a=\left(9a+90\right)a$$

Use the distributive property to multiply 9a+90 by a.

$$5a^{2}+150a=9a^{2}+90a$$

Subtract 9a^{2} from both sides.

$$5a^{2}+150a-9a^{2}=90a$$

Combine 5a^{2} and -9a^{2} to get -4a^{2}.

$$-4a^{2}+150a=90a$$

Subtract 90a from both sides.

$$-4a^{2}+150a-90a=0$$

Combine 150a and -90a to get 60a.

$$-4a^{2}+60a=0$$

Divide both sides by -4.

$$\frac{-4a^{2}+60a}{-4}=\frac{0}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$a^{2}+\frac{60}{-4}a=\frac{0}{-4}$$

Divide 60 by -4.

$$a^{2}-15a=\frac{0}{-4}$$

Divide 0 by -4.

$$a^{2}-15a=0$$

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

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

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-15a+\frac{225}{4}=\frac{225}{4}$$

Factor a^{2}-15a+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{15}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{15}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$a-\frac{15}{2}=\frac{15}{2} a-\frac{15}{2}=-\frac{15}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$a=15 a=0$$