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-16x^{2}+82x-72-6x+24=0
Use the distributive property to multiply -2x+8 by 8x-9 and combine like terms.
-16x^{2}+76x-72+24=0
Combine 82x and -6x to get 76x.
-16x^{2}+76x-48=0
Add -72 and 24 to get -48.
-4x^{2}+19x-12=0
Divide both sides by 4.
a+b=19 ab=-4\left(-12\right)=48
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-12. To find a and b, set up a system to be solved.
1,48 2,24 3,16 4,12 6,8
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 48.
1+48=49 2+24=26 3+16=19 4+12=16 6+8=14
Calculate the sum for each pair.
a=16 b=3
The solution is the pair that gives sum 19.
\left(-4x^{2}+16x\right)+\left(3x-12\right)
Rewrite -4x^{2}+19x-12 as \left(-4x^{2}+16x\right)+\left(3x-12\right).
4x\left(-x+4\right)-3\left(-x+4\right)
Factor out 4x in the first and -3 in the second group.
\left(-x+4\right)\left(4x-3\right)
Factor out common term -x+4 by using distributive property.
x=4 x=\frac{3}{4}
To find equation solutions, solve -x+4=0 and 4x-3=0.
-16x^{2}+82x-72-6x+24=0
Use the distributive property to multiply -2x+8 by 8x-9 and combine like terms.
-16x^{2}+76x-72+24=0
Combine 82x and -6x to get 76x.
-16x^{2}+76x-48=0
Add -72 and 24 to get -48.
x=\frac{-76±\sqrt{76^{2}-4\left(-16\right)\left(-48\right)}}{2\left(-16\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 76 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-76±\sqrt{5776-4\left(-16\right)\left(-48\right)}}{2\left(-16\right)}
Square 76.
x=\frac{-76±\sqrt{5776+64\left(-48\right)}}{2\left(-16\right)}
Multiply -4 times -16.
x=\frac{-76±\sqrt{5776-3072}}{2\left(-16\right)}
Multiply 64 times -48.
x=\frac{-76±\sqrt{2704}}{2\left(-16\right)}
Add 5776 to -3072.
x=\frac{-76±52}{2\left(-16\right)}
Take the square root of 2704.
x=\frac{-76±52}{-32}
Multiply 2 times -16.
x=-\frac{24}{-32}
Now solve the equation x=\frac{-76±52}{-32} when ± is plus. Add -76 to 52.
x=\frac{3}{4}
Reduce the fraction \frac{-24}{-32} to lowest terms by extracting and canceling out 8.
x=-\frac{128}{-32}
Now solve the equation x=\frac{-76±52}{-32} when ± is minus. Subtract 52 from -76.
x=4
Divide -128 by -32.
x=\frac{3}{4} x=4
The equation is now solved.
-16x^{2}+82x-72-6x+24=0
Use the distributive property to multiply -2x+8 by 8x-9 and combine like terms.
-16x^{2}+76x-72+24=0
Combine 82x and -6x to get 76x.
-16x^{2}+76x-48=0
Add -72 and 24 to get -48.
-16x^{2}+76x=48
Add 48 to both sides. Anything plus zero gives itself.
\frac{-16x^{2}+76x}{-16}=\frac{48}{-16}
Divide both sides by -16.
x^{2}+\frac{76}{-16}x=\frac{48}{-16}
Dividing by -16 undoes the multiplication by -16.
x^{2}-\frac{19}{4}x=\frac{48}{-16}
Reduce the fraction \frac{76}{-16} to lowest terms by extracting and canceling out 4.
x^{2}-\frac{19}{4}x=-3
Divide 48 by -16.
x^{2}-\frac{19}{4}x+\left(-\frac{19}{8}\right)^{2}=-3+\left(-\frac{19}{8}\right)^{2}
Divide -\frac{19}{4}, the coefficient of the x term, by 2 to get -\frac{19}{8}. Then add the square of -\frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{19}{4}x+\frac{361}{64}=-3+\frac{361}{64}
Square -\frac{19}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{19}{4}x+\frac{361}{64}=\frac{169}{64}
Add -3 to \frac{361}{64}.
\left(x-\frac{19}{8}\right)^{2}=\frac{169}{64}
Factor x^{2}-\frac{19}{4}x+\frac{361}{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(x-\frac{19}{8}\right)^{2}}=\sqrt{\frac{169}{64}}
Take the square root of both sides of the equation.
x-\frac{19}{8}=\frac{13}{8} x-\frac{19}{8}=-\frac{13}{8}
Simplify.
x=4 x=\frac{3}{4}
Add \frac{19}{8} to both sides of the equation.