Solve 2/b-3-6/2b+1=4

$$\frac{2}{b-3}-\frac{6}{2b+1}=4$$

Solve for b

$$b = \frac{\sqrt{337} + 9}{8} \approx 3.419694969$$

$$b=\frac{9-\sqrt{337}}{8}\approx -1.169694969$$

Steps Using the Quadratic Formula

Variable b cannot be equal to any of the values -\frac{1}{2},3 since division by zero is not defined. Multiply both sides of the equation by \left(b-3\right)\left(2b+1\right), the least common multiple of b-3,2b+1.

$$\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Use the distributive property to multiply 2b+1 by 2.

$$4b+2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Use the distributive property to multiply b-3 by 6.

$$4b+2-\left(6b-18\right)=4\left(b-3\right)\left(2b+1\right)$$

To find the opposite of 6b-18, find the opposite of each term.

$$4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)$$

Combine 4b and -6b to get -2b.

$$-2b+2+18=4\left(b-3\right)\left(2b+1\right)$$

Add 2 and 18 to get 20.

$$-2b+20=4\left(b-3\right)\left(2b+1\right)$$

Use the distributive property to multiply 4 by b-3.

$$-2b+20=\left(4b-12\right)\left(2b+1\right)$$

Use the distributive property to multiply 4b-12 by 2b+1 and combine like terms.

$$-2b+20=8b^{2}-20b-12$$

Subtract 8b^{2} from both sides.

$$-2b+20-8b^{2}=-20b-12$$

Add 20b to both sides.

$$-2b+20-8b^{2}+20b=-12$$

Combine -2b and 20b to get 18b.

$$18b+20-8b^{2}=-12$$

Add 12 to both sides.

$$18b+20-8b^{2}+12=0$$

Add 20 and 12 to get 32.

$$18b+32-8b^{2}=0$$

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.

$$-8b^{2}+18b+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 18 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-18±\sqrt{18^{2}-4\left(-8\right)\times 32}}{2\left(-8\right)}$$

Square 18.

$$b=\frac{-18±\sqrt{324-4\left(-8\right)\times 32}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$b=\frac{-18±\sqrt{324+32\times 32}}{2\left(-8\right)}$$

Multiply 32 times 32.

$$b=\frac{-18±\sqrt{324+1024}}{2\left(-8\right)}$$

Add 324 to 1024.

$$b=\frac{-18±\sqrt{1348}}{2\left(-8\right)}$$

Take the square root of 1348.

$$b=\frac{-18±2\sqrt{337}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$b=\frac{-18±2\sqrt{337}}{-16}$$

Now solve the equation b=\frac{-18±2\sqrt{337}}{-16} when ± is plus. Add -18 to 2\sqrt{337}.

$$b=\frac{2\sqrt{337}-18}{-16}$$

Divide -18+2\sqrt{337} by -16.

$$b=\frac{9-\sqrt{337}}{8}$$

Now solve the equation b=\frac{-18±2\sqrt{337}}{-16} when ± is minus. Subtract 2\sqrt{337} from -18.

$$b=\frac{-2\sqrt{337}-18}{-16}$$

Divide -18-2\sqrt{337} by -16.

$$b=\frac{\sqrt{337}+9}{8}$$

The equation is now solved.

$$b=\frac{9-\sqrt{337}}{8} b=\frac{\sqrt{337}+9}{8}$$

Steps for Completing the Square

Variable b cannot be equal to any of the values -\frac{1}{2},3 since division by zero is not defined. Multiply both sides of the equation by \left(b-3\right)\left(2b+1\right), the least common multiple of b-3,2b+1.

$$\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Use the distributive property to multiply 2b+1 by 2.

$$4b+2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Use the distributive property to multiply b-3 by 6.

$$4b+2-\left(6b-18\right)=4\left(b-3\right)\left(2b+1\right)$$

To find the opposite of 6b-18, find the opposite of each term.

$$4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)$$

Combine 4b and -6b to get -2b.

$$-2b+2+18=4\left(b-3\right)\left(2b+1\right)$$

Add 2 and 18 to get 20.

$$-2b+20=4\left(b-3\right)\left(2b+1\right)$$

Use the distributive property to multiply 4 by b-3.

$$-2b+20=\left(4b-12\right)\left(2b+1\right)$$

Use the distributive property to multiply 4b-12 by 2b+1 and combine like terms.

$$-2b+20=8b^{2}-20b-12$$

Subtract 8b^{2} from both sides.

$$-2b+20-8b^{2}=-20b-12$$

Add 20b to both sides.

$$-2b+20-8b^{2}+20b=-12$$

Combine -2b and 20b to get 18b.

$$18b+20-8b^{2}=-12$$

Subtract 20 from both sides.

$$18b-8b^{2}=-12-20$$

Subtract 20 from -12 to get -32.

$$18b-8b^{2}=-32$$

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.

$$-8b^{2}+18b=-32$$

Divide both sides by -8.

$$\frac{-8b^{2}+18b}{-8}=-\frac{32}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$b^{2}+\frac{18}{-8}b=-\frac{32}{-8}$$

Reduce the fraction \frac{18}{-8} to lowest terms by extracting and canceling out 2.

$$b^{2}-\frac{9}{4}b=-\frac{32}{-8}$$

Divide -32 by -8.

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

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-\frac{9}{4}b+\left(-\frac{9}{8}\right)^{2}=4+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-\frac{9}{4}b+\frac{81}{64}=4+\frac{81}{64}$$

Add 4 to \frac{81}{64}.

$$b^{2}-\frac{9}{4}b+\frac{81}{64}=\frac{337}{64}$$

Factor b^{2}-\frac{9}{4}b+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{9}{8}\right)^{2}=\frac{337}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{9}{8}\right)^{2}}=\sqrt{\frac{337}{64}}$$

Simplify.

$$b-\frac{9}{8}=\frac{\sqrt{337}}{8} b-\frac{9}{8}=-\frac{\sqrt{337}}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$b=\frac{\sqrt{337}+9}{8} b=\frac{9-\sqrt{337}}{8}$$