Solve -4a/a+2=8/a-7

$$- \frac { 4 a } { a + 2 } = \frac { 8 } { a - 7 }$$

Solve for a

$$a=1$$

$$a=4$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(a-7\right)\left(a+2\right), the least common multiple of a+2,a-7.

$$-\left(a-7\right)\times 4a=\left(a+2\right)\times 8$$

Use the distributive property to multiply a-7 by 4.

$$-\left(4a-28\right)a=\left(a+2\right)\times 8$$

Use the distributive property to multiply 4a-28 by a.

$$-\left(4a^{2}-28a\right)=\left(a+2\right)\times 8$$

To find the opposite of 4a^{2}-28a, find the opposite of each term.

$$-4a^{2}+28a=\left(a+2\right)\times 8$$

Use the distributive property to multiply a+2 by 8.

$$-4a^{2}+28a=8a+16$$

Subtract 8a from both sides.

$$-4a^{2}+28a-8a=16$$

Combine 28a and -8a to get 20a.

$$-4a^{2}+20a=16$$

Subtract 16 from both sides.

$$-4a^{2}+20a-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 20 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-20±\sqrt{20^{2}-4\left(-4\right)\left(-16\right)}}{2\left(-4\right)}$$

Square 20.

$$a=\frac{-20±\sqrt{400-4\left(-4\right)\left(-16\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-20±\sqrt{400+16\left(-16\right)}}{2\left(-4\right)}$$

Multiply 16 times -16.

$$a=\frac{-20±\sqrt{400-256}}{2\left(-4\right)}$$

Add 400 to -256.

$$a=\frac{-20±\sqrt{144}}{2\left(-4\right)}$$

Take the square root of 144.

$$a=\frac{-20±12}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{-20±12}{-8}$$

Now solve the equation a=\frac{-20±12}{-8} when ± is plus. Add -20 to 12.

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

Divide -8 by -8.

$$a=1$$

Now solve the equation a=\frac{-20±12}{-8} when ± is minus. Subtract 12 from -20.

$$a=-\frac{32}{-8}$$

Divide -32 by -8.

$$a=4$$

The equation is now solved.

$$a=1 a=4$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(a-7\right)\left(a+2\right), the least common multiple of a+2,a-7.

$$-\left(a-7\right)\times 4a=\left(a+2\right)\times 8$$

Use the distributive property to multiply a-7 by 4.

$$-\left(4a-28\right)a=\left(a+2\right)\times 8$$

Use the distributive property to multiply 4a-28 by a.

$$-\left(4a^{2}-28a\right)=\left(a+2\right)\times 8$$

To find the opposite of 4a^{2}-28a, find the opposite of each term.

$$-4a^{2}+28a=\left(a+2\right)\times 8$$

Use the distributive property to multiply a+2 by 8.

$$-4a^{2}+28a=8a+16$$

Subtract 8a from both sides.

$$-4a^{2}+28a-8a=16$$

Combine 28a and -8a to get 20a.

$$-4a^{2}+20a=16$$

Divide both sides by -4.

$$\frac{-4a^{2}+20a}{-4}=\frac{16}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$a^{2}+\frac{20}{-4}a=\frac{16}{-4}$$

Divide 20 by -4.

$$a^{2}-5a=\frac{16}{-4}$$

Divide 16 by -4.

$$a^{2}-5a=-4$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-5a+\left(-\frac{5}{2}\right)^{2}=-4+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-5a+\frac{25}{4}=-4+\frac{25}{4}$$

Add -4 to \frac{25}{4}.

$$a^{2}-5a+\frac{25}{4}=\frac{9}{4}$$

Factor a^{2}-5a+\frac{25}{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}.

$$\left(a-\frac{5}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{5}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$a-\frac{5}{2}=\frac{3}{2} a-\frac{5}{2}=-\frac{3}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$a=4 a=1$$