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4x^{2}-12x+9-3\left(2x-3\right)+2=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.
4x^{2}-12x+9-6x+9+2=0
Use the distributive property to multiply -3 by 2x-3.
4x^{2}-18x+9+9+2=0
Combine -12x and -6x to get -18x.
4x^{2}-18x+18+2=0
Add 9 and 9 to get 18.
4x^{2}-18x+20=0
Add 18 and 2 to get 20.
2x^{2}-9x+10=0
Divide both sides by 2.
a+b=-9 ab=2\times 10=20
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx+10. To find a and b, set up a system to be solved.
-1,-20 -2,-10 -4,-5
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 20.
-1-20=-21 -2-10=-12 -4-5=-9
Calculate the sum for each pair.
a=-5 b=-4
The solution is the pair that gives sum -9.
\left(2x^{2}-5x\right)+\left(-4x+10\right)
Rewrite 2x^{2}-9x+10 as \left(2x^{2}-5x\right)+\left(-4x+10\right).
x\left(2x-5\right)-2\left(2x-5\right)
Factor out x in the first and -2 in the second group.
\left(2x-5\right)\left(x-2\right)
Factor out common term 2x-5 by using distributive property.
x=\frac{5}{2} x=2
To find equation solutions, solve 2x-5=0 and x-2=0.
4x^{2}-12x+9-3\left(2x-3\right)+2=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.
4x^{2}-12x+9-6x+9+2=0
Use the distributive property to multiply -3 by 2x-3.
4x^{2}-18x+9+9+2=0
Combine -12x and -6x to get -18x.
4x^{2}-18x+18+2=0
Add 9 and 9 to get 18.
4x^{2}-18x+20=0
Add 18 and 2 to get 20.
x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 4\times 20}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -18 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-18\right)±\sqrt{324-4\times 4\times 20}}{2\times 4}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324-16\times 20}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-18\right)±\sqrt{324-320}}{2\times 4}
Multiply -16 times 20.
x=\frac{-\left(-18\right)±\sqrt{4}}{2\times 4}
Add 324 to -320.
x=\frac{-\left(-18\right)±2}{2\times 4}
Take the square root of 4.
x=\frac{18±2}{2\times 4}
The opposite of -18 is 18.
x=\frac{18±2}{8}
Multiply 2 times 4.
x=\frac{20}{8}
Now solve the equation x=\frac{18±2}{8} when ± is plus. Add 18 to 2.
x=\frac{5}{2}
Reduce the fraction \frac{20}{8} to lowest terms by extracting and canceling out 4.
x=\frac{16}{8}
Now solve the equation x=\frac{18±2}{8} when ± is minus. Subtract 2 from 18.
x=2
Divide 16 by 8.
x=\frac{5}{2} x=2
The equation is now solved.
4x^{2}-12x+9-3\left(2x-3\right)+2=0
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.
4x^{2}-12x+9-6x+9+2=0
Use the distributive property to multiply -3 by 2x-3.
4x^{2}-18x+9+9+2=0
Combine -12x and -6x to get -18x.
4x^{2}-18x+18+2=0
Add 9 and 9 to get 18.
4x^{2}-18x+20=0
Add 18 and 2 to get 20.
4x^{2}-18x=-20
Subtract 20 from both sides. Anything subtracted from zero gives its negation.
\frac{4x^{2}-18x}{4}=-\frac{20}{4}
Divide both sides by 4.
x^{2}+\left(-\frac{18}{4}\right)x=-\frac{20}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{9}{2}x=-\frac{20}{4}
Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{9}{2}x=-5
Divide -20 by 4.
x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-5+\left(-\frac{9}{4}\right)^{2}
Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{9}{2}x+\frac{81}{16}=-5+\frac{81}{16}
Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{1}{16}
Add -5 to \frac{81}{16}.
\left(x-\frac{9}{4}\right)^{2}=\frac{1}{16}
Factor x^{2}-\frac{9}{2}x+\frac{81}{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{9}{4}\right)^{2}}=\sqrt{\frac{1}{16}}
Take the square root of both sides of the equation.
x-\frac{9}{4}=\frac{1}{4} x-\frac{9}{4}=-\frac{1}{4}
Simplify.
x=\frac{5}{2} x=2
Add \frac{9}{4} to both sides of the equation.