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2x^{2}-36-x=0
Subtract x from both sides.
2x^{2}-x-36=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-1 ab=2\left(-36\right)=-72
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-36. To find a and b, set up a system to be solved.
1,-72 2,-36 3,-24 4,-18 6,-12 8,-9
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -72.
1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1
Calculate the sum for each pair.
a=-9 b=8
The solution is the pair that gives sum -1.
\left(2x^{2}-9x\right)+\left(8x-36\right)
Rewrite 2x^{2}-x-36 as \left(2x^{2}-9x\right)+\left(8x-36\right).
x\left(2x-9\right)+4\left(2x-9\right)
Factor out x in the first and 4 in the second group.
\left(2x-9\right)\left(x+4\right)
Factor out common term 2x-9 by using distributive property.
x=\frac{9}{2} x=-4
To find equation solutions, solve 2x-9=0 and x+4=0.
2x^{2}-36-x=0
Subtract x from both sides.
2x^{2}-x-36=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.
x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-36\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1-8\left(-36\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 2}
Multiply -8 times -36.
x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 2}
x=\frac{-\left(-1\right)±17}{2\times 2}
Take the square root of 289.
x=\frac{1±17}{2\times 2}
The opposite of -1 is 1.
x=\frac{1±17}{4}
Multiply 2 times 2.
x=\frac{18}{4}
Now solve the equation x=\frac{1±17}{4} when ± is plus. Add 1 to 17.
x=\frac{9}{2}
Reduce the fraction \frac{18}{4}=4.5 to lowest terms by extracting and canceling out 2.
x=\frac{-16}{4}
Now solve the equation x=\frac{1±17}{4} when ± is minus. Subtract 17 from 1.
x=-4
Divide -16 by 4.
x=\frac{9}{2} x=-4
The equation is now solved.
2x^{2}-36-x=0
Subtract x from both sides.
2x^{2}-x=36
Add 36 to both sides. Anything plus zero gives itself.
\frac{2x^{2}-x}{2}=\frac{36}{2}
Divide both sides by 2.
x^{2}+\frac{-1}{2}x=\frac{36}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{1}{2}x=\frac{36}{2}
Divide -1 by 2.
x^{2}-\frac{1}{2}x=18
Divide 36 by 2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=18+\left(-\frac{1}{4}\right)^{2}
Divide -\frac{1}{2}=-0.5, the coefficient of the x term, by 2 to get -\frac{1}{4}=-0.25. Then add the square of -\frac{1}{4}=-0.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{2}x+\frac{1}{16}=18+\frac{1}{16}
Square -\frac{1}{4}=-0.25 by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{289}{16}