Solve b+22/4b-40-b+8/b-10=b/4

$$\frac { b + 22 } { 4 b - 40 } - \frac { b + 8 } { b - 10 } = \frac { b } { 4 }$$

Solve for b

$$b=2$$

$$b=5$$

Steps Using Factoring By Grouping

Variable b cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 4\left(b-10\right), the least common multiple of 4b-40,b-10,4.

$$b+22-4\left(b+8\right)=\left(b-10\right)b$$

Use the distributive property to multiply -4 by b+8.

$$b+22-4b-32=\left(b-10\right)b$$

Combine b and -4b to get -3b.

$$-3b+22-32=\left(b-10\right)b$$

Subtract 32 from 22 to get -10.

$$-3b-10=\left(b-10\right)b$$

Use the distributive property to multiply b-10 by b.

$$-3b-10=b^{2}-10b$$

Subtract b^{2} from both sides.

$$-3b-10-b^{2}=-10b$$

Add 10b to both sides.

$$-3b-10-b^{2}+10b=0$$

Combine -3b and 10b to get 7b.

$$7b-10-b^{2}=0$$

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

$$-b^{2}+7b-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -b^{2}+ab+bb-10. To find a and b, set up a system to be solved.

$$a+b=7 ab=-\left(-10\right)=10$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 7.

$$a=5 b=2$$

Rewrite -b^{2}+7b-10 as \left(-b^{2}+5b\right)+\left(2b-10\right).

$$\left(-b^{2}+5b\right)+\left(2b-10\right)$$

Factor out -b in the first and 2 in the second group.

$$-b\left(b-5\right)+2\left(b-5\right)$$

Factor out common term b-5 by using distributive property.

$$\left(b-5\right)\left(-b+2\right)$$

To find equation solutions, solve b-5=0 and -b+2=0.

$$b=5 b=2$$

Steps Using the Quadratic Formula

Variable b cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 4\left(b-10\right), the least common multiple of 4b-40,b-10,4.

$$b+22-4\left(b+8\right)=\left(b-10\right)b$$

Use the distributive property to multiply -4 by b+8.

$$b+22-4b-32=\left(b-10\right)b$$

Combine b and -4b to get -3b.

$$-3b+22-32=\left(b-10\right)b$$

Subtract 32 from 22 to get -10.

$$-3b-10=\left(b-10\right)b$$

Use the distributive property to multiply b-10 by b.

$$-3b-10=b^{2}-10b$$

Subtract b^{2} from both sides.

$$-3b-10-b^{2}=-10b$$

Add 10b to both sides.

$$-3b-10-b^{2}+10b=0$$

Combine -3b and 10b to get 7b.

$$7b-10-b^{2}=0$$

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.

$$-b^{2}+7b-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Square 7.

$$b=\frac{-7±\sqrt{49-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-7±\sqrt{49+4\left(-10\right)}}{2\left(-1\right)}$$

Multiply 4 times -10.

$$b=\frac{-7±\sqrt{49-40}}{2\left(-1\right)}$$

Add 49 to -40.

$$b=\frac{-7±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

$$b=\frac{-7±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-7±3}{-2}$$

Now solve the equation b=\frac{-7±3}{-2} when ± is plus. Add -7 to 3.

$$b=-\frac{4}{-2}$$

Divide -4 by -2.

$$b=2$$

Now solve the equation b=\frac{-7±3}{-2} when ± is minus. Subtract 3 from -7.

$$b=-\frac{10}{-2}$$

Divide -10 by -2.

$$b=5$$

The equation is now solved.

$$b=2 b=5$$

Steps for Completing the Square

Variable b cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 4\left(b-10\right), the least common multiple of 4b-40,b-10,4.

$$b+22-4\left(b+8\right)=\left(b-10\right)b$$

Use the distributive property to multiply -4 by b+8.

$$b+22-4b-32=\left(b-10\right)b$$

Combine b and -4b to get -3b.

$$-3b+22-32=\left(b-10\right)b$$

Subtract 32 from 22 to get -10.

$$-3b-10=\left(b-10\right)b$$

Use the distributive property to multiply b-10 by b.

$$-3b-10=b^{2}-10b$$

Subtract b^{2} from both sides.

$$-3b-10-b^{2}=-10b$$

Add 10b to both sides.

$$-3b-10-b^{2}+10b=0$$

Combine -3b and 10b to get 7b.

$$7b-10-b^{2}=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$7b-b^{2}=10$$

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.

$$-b^{2}+7b=10$$

Divide both sides by -1.

$$\frac{-b^{2}+7b}{-1}=\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b^{2}+\frac{7}{-1}b=\frac{10}{-1}$$

Divide 7 by -1.

$$b^{2}-7b=\frac{10}{-1}$$

Divide 10 by -1.

$$b^{2}-7b=-10$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-7b+\left(-\frac{7}{2}\right)^{2}=-10+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-7b+\frac{49}{4}=-10+\frac{49}{4}$$

Add -10 to \frac{49}{4}.

$$b^{2}-7b+\frac{49}{4}=\frac{9}{4}$$

Factor b^{2}-7b+\frac{49}{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}.

$$\left(b-\frac{7}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{7}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$b-\frac{7}{2}=\frac{3}{2} b-\frac{7}{2}=-\frac{3}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$b=5 b=2$$