Solve 8/a^2-25-2/a-5=1/29+10

$$\frac { 8 } { a ^ { 2 } - 25 } - \frac { 2 } { a - 5 } = \frac { 1 } { 29 + 10 }$$

Solve for a

$$a=2\sqrt{367}-39\approx -0.685511879$$

$$a=-2\sqrt{367}-39\approx -77.314488121$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(a-5\right)\left(a+5\right), the least common multiple of a^{2}-25,a-5.

$$8-\left(a+5\right)\times 2=\frac{1}{39}a^{2}-\frac{25}{39}$$

Use the distributive property to multiply a+5 by 2.

$$8-\left(2a+10\right)=\frac{1}{39}a^{2}-\frac{25}{39}$$

To find the opposite of 2a+10, find the opposite of each term.

$$8-2a-10=\frac{1}{39}a^{2}-\frac{25}{39}$$

Subtract 10 from 8 to get -2.

$$-2-2a=\frac{1}{39}a^{2}-\frac{25}{39}$$

Subtract \frac{1}{39}a^{2} from both sides.

$$-2-2a-\frac{1}{39}a^{2}=-\frac{25}{39}$$

Add \frac{25}{39} to both sides.

$$-2-2a-\frac{1}{39}a^{2}+\frac{25}{39}=0$$

Add -2 and \frac{25}{39} to get -\frac{53}{39}.

$$-\frac{53}{39}-2a-\frac{1}{39}a^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-\frac{1}{39}a^{2}-2a-\frac{53}{39}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{39} for a, -2 for b, and -\frac{53}{39} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-\frac{1}{39}\right)\left(-\frac{53}{39}\right)}}{2\left(-\frac{1}{39}\right)}$$

Square -2.

$$a=\frac{-\left(-2\right)±\sqrt{4-4\left(-\frac{1}{39}\right)\left(-\frac{53}{39}\right)}}{2\left(-\frac{1}{39}\right)}$$

Multiply -4 times -\frac{1}{39}.

$$a=\frac{-\left(-2\right)±\sqrt{4+\frac{4}{39}\left(-\frac{53}{39}\right)}}{2\left(-\frac{1}{39}\right)}$$

Multiply \frac{4}{39} times -\frac{53}{39} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$a=\frac{-\left(-2\right)±\sqrt{4-\frac{212}{1521}}}{2\left(-\frac{1}{39}\right)}$$

Add 4 to -\frac{212}{1521}.

$$a=\frac{-\left(-2\right)±\sqrt{\frac{5872}{1521}}}{2\left(-\frac{1}{39}\right)}$$

Take the square root of \frac{5872}{1521}.

$$a=\frac{-\left(-2\right)±\frac{4\sqrt{367}}{39}}{2\left(-\frac{1}{39}\right)}$$

The opposite of -2 is 2.

$$a=\frac{2±\frac{4\sqrt{367}}{39}}{2\left(-\frac{1}{39}\right)}$$

Multiply 2 times -\frac{1}{39}.

$$a=\frac{2±\frac{4\sqrt{367}}{39}}{-\frac{2}{39}}$$

Now solve the equation a=\frac{2±\frac{4\sqrt{367}}{39}}{-\frac{2}{39}} when ± is plus. Add 2 to \frac{4\sqrt{367}}{39}.

$$a=\frac{\frac{4\sqrt{367}}{39}+2}{-\frac{2}{39}}$$

Divide 2+\frac{4\sqrt{367}}{39} by -\frac{2}{39} by multiplying 2+\frac{4\sqrt{367}}{39} by the reciprocal of -\frac{2}{39}.

$$a=-2\sqrt{367}-39$$

Now solve the equation a=\frac{2±\frac{4\sqrt{367}}{39}}{-\frac{2}{39}} when ± is minus. Subtract \frac{4\sqrt{367}}{39} from 2.

$$a=\frac{-\frac{4\sqrt{367}}{39}+2}{-\frac{2}{39}}$$

Divide 2-\frac{4\sqrt{367}}{39} by -\frac{2}{39} by multiplying 2-\frac{4\sqrt{367}}{39} by the reciprocal of -\frac{2}{39}.

$$a=2\sqrt{367}-39$$

The equation is now solved.

$$a=-2\sqrt{367}-39 a=2\sqrt{367}-39$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(a-5\right)\left(a+5\right), the least common multiple of a^{2}-25,a-5.

$$8-\left(a+5\right)\times 2=\frac{1}{39}a^{2}-\frac{25}{39}$$

Use the distributive property to multiply a+5 by 2.

$$8-\left(2a+10\right)=\frac{1}{39}a^{2}-\frac{25}{39}$$

To find the opposite of 2a+10, find the opposite of each term.

$$8-2a-10=\frac{1}{39}a^{2}-\frac{25}{39}$$

Subtract 10 from 8 to get -2.

$$-2-2a=\frac{1}{39}a^{2}-\frac{25}{39}$$

Subtract \frac{1}{39}a^{2} from both sides.

$$-2-2a-\frac{1}{39}a^{2}=-\frac{25}{39}$$

Add 2 to both sides.

$$-2a-\frac{1}{39}a^{2}=-\frac{25}{39}+2$$

Add -\frac{25}{39} and 2 to get \frac{53}{39}.

$$-2a-\frac{1}{39}a^{2}=\frac{53}{39}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-\frac{1}{39}a^{2}-2a=\frac{53}{39}$$

Multiply both sides by -39.

$$\frac{-\frac{1}{39}a^{2}-2a}{-\frac{1}{39}}=\frac{\frac{53}{39}}{-\frac{1}{39}}$$

Dividing by -\frac{1}{39} undoes the multiplication by -\frac{1}{39}.

$$a^{2}+\left(-\frac{2}{-\frac{1}{39}}\right)a=\frac{\frac{53}{39}}{-\frac{1}{39}}$$

Divide -2 by -\frac{1}{39} by multiplying -2 by the reciprocal of -\frac{1}{39}.

$$a^{2}+78a=\frac{\frac{53}{39}}{-\frac{1}{39}}$$

Divide \frac{53}{39} by -\frac{1}{39} by multiplying \frac{53}{39} by the reciprocal of -\frac{1}{39}.

$$a^{2}+78a=-53$$

Divide 78, the coefficient of the x term, by 2 to get 39. Then add the square of 39 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+78a+39^{2}=-53+39^{2}$$

Square 39.

$$a^{2}+78a+1521=-53+1521$$

Add -53 to 1521.

$$a^{2}+78a+1521=1468$$

Factor a^{2}+78a+1521. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+39\right)^{2}=1468$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+39\right)^{2}}=\sqrt{1468}$$

Simplify.

$$a+39=2\sqrt{367} a+39=-2\sqrt{367}$$

Subtract 39 from both sides of the equation.

$$a=2\sqrt{367}-39 a=-2\sqrt{367}-39$$