Solve for b
b = \frac{\sqrt{337} + 9}{8} \approx 3.419694969
b=\frac{9-\sqrt{337}}{8}\approx -1.169694969
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\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)
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.
4b+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(6b-18\right)=4\left(b-3\right)\left(2b+1\right)
Use the distributive property to multiply b-3 by 6.
4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)
To find the opposite of 6b-18, find the opposite of each term.
-2b+2+18=4\left(b-3\right)\left(2b+1\right)
Combine 4b and -6b to get -2b.
-2b+20=4\left(b-3\right)\left(2b+1\right)
Add 2 and 18 to get 20.
-2b+20=\left(4b-12\right)\left(2b+1\right)
Use the distributive property to multiply 4 by b-3.
-2b+20=8b^{2}-20b-12
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.
18b+20-8b^{2}=-12
Combine -2b and 20b to get 18b.
18b+20-8b^{2}+12=0
Add 12 to both sides.
18b+32-8b^{2}=0
Add 20 and 12 to get 32.
-8b^{2}+18b+32=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=\frac{-18±\sqrt{18^{2}-4\left(-8\right)\times 32}}{2\left(-8\right)}
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{324-4\left(-8\right)\times 32}}{2\left(-8\right)}
Square 18.
b=\frac{-18±\sqrt{324+32\times 32}}{2\left(-8\right)}
Multiply -4 times -8.
b=\frac{-18±\sqrt{324+1024}}{2\left(-8\right)}
Multiply 32 times 32.
b=\frac{-18±\sqrt{1348}}{2\left(-8\right)}
Add 324 to 1024.
b=\frac{-18±2\sqrt{337}}{2\left(-8\right)}
Take the square root of 1348.
b=\frac{-18±2\sqrt{337}}{-16}
Multiply 2 times -8.
b=\frac{2\sqrt{337}-18}{-16}
Now solve the equation b=\frac{-18±2\sqrt{337}}{-16} when ± is plus. Add -18 to 2\sqrt{337}.
b=\frac{9-\sqrt{337}}{8}
Divide -18+2\sqrt{337} by -16.
b=\frac{-2\sqrt{337}-18}{-16}
Now solve the equation b=\frac{-18±2\sqrt{337}}{-16} when ± is minus. Subtract 2\sqrt{337} from -18.
b=\frac{\sqrt{337}+9}{8}
Divide -18-2\sqrt{337} by -16.
b=\frac{9-\sqrt{337}}{8} b=\frac{\sqrt{337}+9}{8}
The equation is now solved.
\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)
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.
4b+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(6b-18\right)=4\left(b-3\right)\left(2b+1\right)
Use the distributive property to multiply b-3 by 6.
4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)
To find the opposite of 6b-18, find the opposite of each term.
-2b+2+18=4\left(b-3\right)\left(2b+1\right)
Combine 4b and -6b to get -2b.
-2b+20=4\left(b-3\right)\left(2b+1\right)
Add 2 and 18 to get 20.
-2b+20=\left(4b-12\right)\left(2b+1\right)
Use the distributive property to multiply 4 by b-3.
-2b+20=8b^{2}-20b-12
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.
18b+20-8b^{2}=-12
Combine -2b and 20b to get 18b.
18b-8b^{2}=-12-20
Subtract 20 from both sides.
18b-8b^{2}=-32
Subtract 20 from -12 to get -32.
-8b^{2}+18b=-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.
\frac{-8b^{2}+18b}{-8}=-\frac{32}{-8}
Divide both sides by -8.
b^{2}+\frac{18}{-8}b=-\frac{32}{-8}
Dividing by -8 undoes the multiplication by -8.
b^{2}-\frac{9}{4}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=4
Divide -32 by -8.
b^{2}-\frac{9}{4}b+\left(-\frac{9}{8}\right)^{2}=4+\left(-\frac{9}{8}\right)^{2}
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+\frac{81}{64}=4+\frac{81}{64}
Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.
b^{2}-\frac{9}{4}b+\frac{81}{64}=\frac{337}{64}
Add 4 to \frac{81}{64}.
\left(b-\frac{9}{8}\right)^{2}=\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}.
\sqrt{\left(b-\frac{9}{8}\right)^{2}}=\sqrt{\frac{337}{64}}
Take the square root of both sides of the equation.
b-\frac{9}{8}=\frac{\sqrt{337}}{8} b-\frac{9}{8}=-\frac{\sqrt{337}}{8}
Simplify.
b=\frac{\sqrt{337}+9}{8} b=\frac{9-\sqrt{337}}{8}
Add \frac{9}{8} to both sides of the equation.