Solve a/10=3/a+7

$$\frac { a } { 10 } = \frac { 3 } { a + 7 }$$

Solve for a

$$a=-10$$

$$a=3$$

Steps Using the Quadratic Formula

Variable a cannot be equal to -7 since division by zero is not defined. Multiply both sides of the equation by 10\left(a+7\right), the least common multiple of 10,a+7.

$$\left(a+7\right)a=10\times 3$$

Use the distributive property to multiply a+7 by a.

$$a^{2}+7a=10\times 3$$

Multiply 10 and 3 to get 30.

$$a^{2}+7a=30$$

Subtract 30 from both sides.

$$a^{2}+7a-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-7±\sqrt{7^{2}-4\left(-30\right)}}{2}$$

Square 7.

$$a=\frac{-7±\sqrt{49-4\left(-30\right)}}{2}$$

Multiply -4 times -30.

$$a=\frac{-7±\sqrt{49+120}}{2}$$

Add 49 to 120.

$$a=\frac{-7±\sqrt{169}}{2}$$

Take the square root of 169.

$$a=\frac{-7±13}{2}$$

Now solve the equation a=\frac{-7±13}{2} when ± is plus. Add -7 to 13.

$$a=\frac{6}{2}$$

Divide 6 by 2.

$$a=3$$

Now solve the equation a=\frac{-7±13}{2} when ± is minus. Subtract 13 from -7.

$$a=-\frac{20}{2}$$

Divide -20 by 2.

$$a=-10$$

The equation is now solved.

$$a=3 a=-10$$

Steps for Completing the Square

Variable a cannot be equal to -7 since division by zero is not defined. Multiply both sides of the equation by 10\left(a+7\right), the least common multiple of 10,a+7.

$$\left(a+7\right)a=10\times 3$$

Use the distributive property to multiply a+7 by a.

$$a^{2}+7a=10\times 3$$

Multiply 10 and 3 to get 30.

$$a^{2}+7a=30$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+7a+\left(\frac{7}{2}\right)^{2}=30+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+7a+\frac{49}{4}=30+\frac{49}{4}$$

Add 30 to \frac{49}{4}.

$$a^{2}+7a+\frac{49}{4}=\frac{169}{4}$$

Factor a^{2}+7a+\frac{49}{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}.

$$\left(a+\frac{7}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{7}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$a+\frac{7}{2}=\frac{13}{2} a+\frac{7}{2}=-\frac{13}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$a=3 a=-10$$