Solve -2a/a+25=4/a-17

$$- \frac { 2 a } { a + 25 } = \frac { 4 } { a - 17 }$$

Solve for a

$$a=5$$

$$a=10$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -25,17 since division by zero is not defined. Multiply both sides of the equation by \left(a-17\right)\left(a+25\right), the least common multiple of a+25,a-17.

$$-\left(a-17\right)\times 2a=\left(a+25\right)\times 4$$

Use the distributive property to multiply a-17 by 2.

$$-\left(2a-34\right)a=\left(a+25\right)\times 4$$

Use the distributive property to multiply 2a-34 by a.

$$-\left(2a^{2}-34a\right)=\left(a+25\right)\times 4$$

To find the opposite of 2a^{2}-34a, find the opposite of each term.

$$-2a^{2}+34a=\left(a+25\right)\times 4$$

Use the distributive property to multiply a+25 by 4.

$$-2a^{2}+34a=4a+100$$

Subtract 4a from both sides.

$$-2a^{2}+34a-4a=100$$

Combine 34a and -4a to get 30a.

$$-2a^{2}+30a=100$$

Subtract 100 from both sides.

$$-2a^{2}+30a-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 30 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-30±\sqrt{30^{2}-4\left(-2\right)\left(-100\right)}}{2\left(-2\right)}$$

Square 30.

$$a=\frac{-30±\sqrt{900-4\left(-2\right)\left(-100\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$a=\frac{-30±\sqrt{900+8\left(-100\right)}}{2\left(-2\right)}$$

Multiply 8 times -100.

$$a=\frac{-30±\sqrt{900-800}}{2\left(-2\right)}$$

Add 900 to -800.

$$a=\frac{-30±\sqrt{100}}{2\left(-2\right)}$$

Take the square root of 100.

$$a=\frac{-30±10}{2\left(-2\right)}$$

Multiply 2 times -2.

$$a=\frac{-30±10}{-4}$$

Now solve the equation a=\frac{-30±10}{-4} when ± is plus. Add -30 to 10.

$$a=-\frac{20}{-4}$$

Divide -20 by -4.

$$a=5$$

Now solve the equation a=\frac{-30±10}{-4} when ± is minus. Subtract 10 from -30.

$$a=-\frac{40}{-4}$$

Divide -40 by -4.

$$a=10$$

The equation is now solved.

$$a=5 a=10$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -25,17 since division by zero is not defined. Multiply both sides of the equation by \left(a-17\right)\left(a+25\right), the least common multiple of a+25,a-17.

$$-\left(a-17\right)\times 2a=\left(a+25\right)\times 4$$

Use the distributive property to multiply a-17 by 2.

$$-\left(2a-34\right)a=\left(a+25\right)\times 4$$

Use the distributive property to multiply 2a-34 by a.

$$-\left(2a^{2}-34a\right)=\left(a+25\right)\times 4$$

To find the opposite of 2a^{2}-34a, find the opposite of each term.

$$-2a^{2}+34a=\left(a+25\right)\times 4$$

Use the distributive property to multiply a+25 by 4.

$$-2a^{2}+34a=4a+100$$

Subtract 4a from both sides.

$$-2a^{2}+34a-4a=100$$

Combine 34a and -4a to get 30a.

$$-2a^{2}+30a=100$$

Divide both sides by -2.

$$\frac{-2a^{2}+30a}{-2}=\frac{100}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$a^{2}+\frac{30}{-2}a=\frac{100}{-2}$$

Divide 30 by -2.

$$a^{2}-15a=\frac{100}{-2}$$

Divide 100 by -2.

$$a^{2}-15a=-50$$

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.

$$a^{2}-15a+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

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

$$a^{2}-15a+\frac{225}{4}=-50+\frac{225}{4}$$

Add -50 to \frac{225}{4}.

$$a^{2}-15a+\frac{225}{4}=\frac{25}{4}$$

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}.

$$\left(a-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$a-\frac{15}{2}=\frac{5}{2} a-\frac{15}{2}=-\frac{5}{2}$$

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

$$a=10 a=5$$