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2\left(x^{2}-8x+16\right)=3\left(x-4\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.
2x^{2}-16x+32=3\left(x-4\right)
Use the distributive property to multiply 2 by x^{2}-8x+16.
2x^{2}-16x+32=3x-12
Use the distributive property to multiply 3 by x-4.
2x^{2}-16x+32-3x=-12
Subtract 3x from both sides.
2x^{2}-19x+32=-12
Combine -16x and -3x to get -19x.
2x^{2}-19x+32+12=0
Add 12 to both sides.
2x^{2}-19x+44=0
Add 32 and 12 to get 44.
a+b=-19 ab=2\times 44=88
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx+44. To find a and b, set up a system to be solved.
-1,-88 -2,-44 -4,-22 -8,-11
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 88.
-1-88=-89 -2-44=-46 -4-22=-26 -8-11=-19
Calculate the sum for each pair.
a=-11 b=-8
The solution is the pair that gives sum -19.
\left(2x^{2}-11x\right)+\left(-8x+44\right)
Rewrite 2x^{2}-19x+44 as \left(2x^{2}-11x\right)+\left(-8x+44\right).
x\left(2x-11\right)-4\left(2x-11\right)
Factor out x in the first and -4 in the second group.
\left(2x-11\right)\left(x-4\right)
Factor out common term 2x-11 by using distributive property.
x=\frac{11}{2} x=4
To find equation solutions, solve 2x-11=0 and x-4=0.
2\left(x^{2}-8x+16\right)=3\left(x-4\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.
2x^{2}-16x+32=3\left(x-4\right)
Use the distributive property to multiply 2 by x^{2}-8x+16.
2x^{2}-16x+32=3x-12
Use the distributive property to multiply 3 by x-4.
2x^{2}-16x+32-3x=-12
Subtract 3x from both sides.
2x^{2}-19x+32=-12
Combine -16x and -3x to get -19x.
2x^{2}-19x+32+12=0
Add 12 to both sides.
2x^{2}-19x+44=0
Add 32 and 12 to get 44.
x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 2\times 44}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -19 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-19\right)±\sqrt{361-4\times 2\times 44}}{2\times 2}
Square -19.
x=\frac{-\left(-19\right)±\sqrt{361-8\times 44}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-19\right)±\sqrt{361-352}}{2\times 2}
Multiply -8 times 44.
x=\frac{-\left(-19\right)±\sqrt{9}}{2\times 2}
Add 361 to -352.
x=\frac{-\left(-19\right)±3}{2\times 2}
Take the square root of 9.
x=\frac{19±3}{2\times 2}
The opposite of -19 is 19.
x=\frac{19±3}{4}
Multiply 2 times 2.
x=\frac{22}{4}
Now solve the equation x=\frac{19±3}{4} when ± is plus. Add 19 to 3.
x=\frac{11}{2}
Reduce the fraction \frac{22}{4} to lowest terms by extracting and canceling out 2.
x=\frac{16}{4}
Now solve the equation x=\frac{19±3}{4} when ± is minus. Subtract 3 from 19.
x=4
Divide 16 by 4.
x=\frac{11}{2} x=4
The equation is now solved.
2\left(x^{2}-8x+16\right)=3\left(x-4\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.
2x^{2}-16x+32=3\left(x-4\right)
Use the distributive property to multiply 2 by x^{2}-8x+16.
2x^{2}-16x+32=3x-12
Use the distributive property to multiply 3 by x-4.
2x^{2}-16x+32-3x=-12
Subtract 3x from both sides.
2x^{2}-19x+32=-12
Combine -16x and -3x to get -19x.
2x^{2}-19x=-12-32
Subtract 32 from both sides.
2x^{2}-19x=-44
Subtract 32 from -12 to get -44.
\frac{2x^{2}-19x}{2}=-\frac{44}{2}
Divide both sides by 2.
x^{2}-\frac{19}{2}x=-\frac{44}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{19}{2}x=-22
Divide -44 by 2.
x^{2}-\frac{19}{2}x+\left(-\frac{19}{4}\right)^{2}=-22+\left(-\frac{19}{4}\right)^{2}
Divide -\frac{19}{2}, the coefficient of the x term, by 2 to get -\frac{19}{4}. Then add the square of -\frac{19}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{19}{2}x+\frac{361}{16}=-22+\frac{361}{16}
Square -\frac{19}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{19}{2}x+\frac{361}{16}=\frac{9}{16}
Add -22 to \frac{361}{16}.
\left(x-\frac{19}{4}\right)^{2}=\frac{9}{16}
Factor x^{2}-\frac{19}{2}x+\frac{361}{16}. 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}{4}\right)^{2}}=\sqrt{\frac{9}{16}}
Take the square root of both sides of the equation.
x-\frac{19}{4}=\frac{3}{4} x-\frac{19}{4}=-\frac{3}{4}
Simplify.
x=\frac{11}{2} x=4
Add \frac{19}{4} to both sides of the equation.