Solve 144/b^2=1+18/b

$$\frac { 144 } { b ^ { 2 } } = 1 + \frac { 18 } { b }$$

Solve for b

$$b=-24$$

$$b=6$$

Steps Using Factoring

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$144=b^{2}+b\times 18$$

Swap sides so that all variable terms are on the left hand side.

$$b^{2}+b\times 18=144$$

Subtract 144 from both sides.

$$b^{2}+b\times 18-144=0$$

To solve the equation, factor b^{2}+18b-144 using formula b^{2}+\left(a+b\right)b+ab=\left(b+a\right)\left(b+b\right). To find a and b, set up a system to be solved.

$$a+b=18 ab=-144$$

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 -144.

$$-1,144 -2,72 -3,48 -4,36 -6,24 -8,18 -9,16 -12,12$$

Calculate the sum for each pair.

$$-1+144=143 -2+72=70 -3+48=45 -4+36=32 -6+24=18 -8+18=10 -9+16=7 -12+12=0$$

The solution is the pair that gives sum 18.

$$a=-6 b=24$$

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

$$\left(b-6\right)\left(b+24\right)$$

To find equation solutions, solve b-6=0 and b+24=0.

$$b=6 b=-24$$

Steps Using Factoring By Grouping

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$144=b^{2}+b\times 18$$

Swap sides so that all variable terms are on the left hand side.

$$b^{2}+b\times 18=144$$

Subtract 144 from both sides.

$$b^{2}+b\times 18-144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as b^{2}+ab+bb-144. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\left(-144\right)=-144$$

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 -144.

$$-1,144 -2,72 -3,48 -4,36 -6,24 -8,18 -9,16 -12,12$$

Calculate the sum for each pair.

$$-1+144=143 -2+72=70 -3+48=45 -4+36=32 -6+24=18 -8+18=10 -9+16=7 -12+12=0$$

The solution is the pair that gives sum 18.

$$a=-6 b=24$$

Rewrite b^{2}+18b-144 as \left(b^{2}-6b\right)+\left(24b-144\right).

$$\left(b^{2}-6b\right)+\left(24b-144\right)$$

Factor out b in the first and 24 in the second group.

$$b\left(b-6\right)+24\left(b-6\right)$$

Factor out common term b-6 by using distributive property.

$$\left(b-6\right)\left(b+24\right)$$

To find equation solutions, solve b-6=0 and b+24=0.

$$b=6 b=-24$$

Steps Using the Quadratic Formula

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$144=b^{2}+b\times 18$$

Swap sides so that all variable terms are on the left hand side.

$$b^{2}+b\times 18=144$$

Subtract 144 from both sides.

$$b^{2}+b\times 18-144=0$$

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.

$$b^{2}+18b-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-18±\sqrt{18^{2}-4\left(-144\right)}}{2}$$

Square 18.

$$b=\frac{-18±\sqrt{324-4\left(-144\right)}}{2}$$

Multiply -4 times -144.

$$b=\frac{-18±\sqrt{324+576}}{2}$$

Add 324 to 576.

$$b=\frac{-18±\sqrt{900}}{2}$$

Take the square root of 900.

$$b=\frac{-18±30}{2}$$

Now solve the equation b=\frac{-18±30}{2} when ± is plus. Add -18 to 30.

$$b=\frac{12}{2}$$

Divide 12 by 2.

$$b=6$$

Now solve the equation b=\frac{-18±30}{2} when ± is minus. Subtract 30 from -18.

$$b=-\frac{48}{2}$$

Divide -48 by 2.

$$b=-24$$

The equation is now solved.

$$b=6 b=-24$$

Steps for Completing the Square

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$144=b^{2}+b\times 18$$

Swap sides so that all variable terms are on the left hand side.

$$b^{2}+b\times 18=144$$

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.

$$b^{2}+18b=144$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}+18b+9^{2}=144+9^{2}$$

Square 9.

$$b^{2}+18b+81=144+81$$

Add 144 to 81.

$$b^{2}+18b+81=225$$

Factor b^{2}+18b+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b+9\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b+9\right)^{2}}=\sqrt{225}$$

Simplify.

$$b+9=15 b+9=-15$$

Subtract 9 from both sides of the equation.

$$b=6 b=-24$$