Solve a-9/a=-8

$$a - \frac { 9 } { a } = - 8$$

Solve for a

$$a=-9$$

$$a=1$$

Steps Using Factoring

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$aa-9=-8a$$

Multiply a and a to get a^{2}.

$$a^{2}-9=-8a$$

Add 8a to both sides.

$$a^{2}-9+8a=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$a^{2}+8a-9=0$$

To solve the equation, factor a^{2}+8a-9 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.

$$a+b=8 ab=-9$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -9.

$$-1,9 -3,3$$

Calculate the sum for each pair.

$$-1+9=8 -3+3=0$$

The solution is the pair that gives sum 8.

$$a=-1 b=9$$

Rewrite factored expression \left(a+a\right)\left(a+b\right) using the obtained values.

$$\left(a-1\right)\left(a+9\right)$$

To find equation solutions, solve a-1=0 and a+9=0.

$$a=1 a=-9$$

Steps Using Factoring By Grouping

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$aa-9=-8a$$

Multiply a and a to get a^{2}.

$$a^{2}-9=-8a$$

Add 8a to both sides.

$$a^{2}-9+8a=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$a^{2}+8a-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba-9. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\left(-9\right)=-9$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -9.

$$-1,9 -3,3$$

Calculate the sum for each pair.

$$-1+9=8 -3+3=0$$

The solution is the pair that gives sum 8.

$$a=-1 b=9$$

Rewrite a^{2}+8a-9 as \left(a^{2}-a\right)+\left(9a-9\right).

$$\left(a^{2}-a\right)+\left(9a-9\right)$$

Factor out a in the first and 9 in the second group.

$$a\left(a-1\right)+9\left(a-1\right)$$

Factor out common term a-1 by using distributive property.

$$\left(a-1\right)\left(a+9\right)$$

To find equation solutions, solve a-1=0 and a+9=0.

$$a=1 a=-9$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$aa-9=-8a$$

Multiply a and a to get a^{2}.

$$a^{2}-9=-8a$$

Add 8a to both sides.

$$a^{2}-9+8a=0$$

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.

$$a^{2}+8a-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-8±\sqrt{8^{2}-4\left(-9\right)}}{2}$$

Square 8.

$$a=\frac{-8±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$a=\frac{-8±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$a=\frac{-8±\sqrt{100}}{2}$$

Take the square root of 100.

$$a=\frac{-8±10}{2}$$

Now solve the equation a=\frac{-8±10}{2} when ± is plus. Add -8 to 10.

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

Divide 2 by 2.

$$a=1$$

Now solve the equation a=\frac{-8±10}{2} when ± is minus. Subtract 10 from -8.

$$a=-\frac{18}{2}$$

Divide -18 by 2.

$$a=-9$$

The equation is now solved.

$$a=1 a=-9$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$aa-9=-8a$$

Multiply a and a to get a^{2}.

$$a^{2}-9=-8a$$

Add 8a to both sides.

$$a^{2}-9+8a=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$a^{2}+8a=9$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+8a+4^{2}=9+4^{2}$$

Square 4.

$$a^{2}+8a+16=9+16$$

Add 9 to 16.

$$a^{2}+8a+16=25$$

Factor a^{2}+8a+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(a+4\right)^{2}}=\sqrt{25}$$

Simplify.

$$a+4=5 a+4=-5$$

Subtract 4 from both sides of the equation.

$$a=1 a=-9$$