Solve for x
x = \frac{3 \sqrt{33} + 3}{4} \approx 5.058421985
x=\frac{3-3\sqrt{33}}{4}\approx -3.558421985
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2x^{2}-3x=36
Subtract 3x from both sides.
2x^{2}-3x-36=0
Subtract 36 from both sides.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-36\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-36\right)}}{2\times 2}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-8\left(-36\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-3\right)±\sqrt{9+288}}{2\times 2}
Multiply -8 times -36.
x=\frac{-\left(-3\right)±\sqrt{297}}{2\times 2}
Add 9 to 288.
x=\frac{-\left(-3\right)±3\sqrt{33}}{2\times 2}
Take the square root of 297.
x=\frac{3±3\sqrt{33}}{2\times 2}
The opposite of -3 is 3.
x=\frac{3±3\sqrt{33}}{4}
Multiply 2 times 2.
x=\frac{3\sqrt{33}+3}{4}
Now solve the equation x=\frac{3±3\sqrt{33}}{4} when ± is plus. Add 3 to 3\sqrt{33}.
x=\frac{3-3\sqrt{33}}{4}
Now solve the equation x=\frac{3±3\sqrt{33}}{4} when ± is minus. Subtract 3\sqrt{33} from 3.
x=\frac{3\sqrt{33}+3}{4} x=\frac{3-3\sqrt{33}}{4}
The equation is now solved.
2x^{2}-3x=36
Subtract 3x from both sides.
\frac{2x^{2}-3x}{2}=\frac{36}{2}
Divide both sides by 2.
x^{2}-\frac{3}{2}x=\frac{36}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{3}{2}x=18
Divide 36 by 2.
x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=18+\left(-\frac{3}{4}\right)^{2}
Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{3}{2}x+\frac{9}{16}=18+\frac{9}{16}
Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{297}{16}
Add 18 to \frac{9}{16}.
\left(x-\frac{3}{4}\right)^{2}=\frac{297}{16}
Factor x^{2}-\frac{3}{2}x+\frac{9}{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{3}{4}\right)^{2}}=\sqrt{\frac{297}{16}}
Take the square root of both sides of the equation.
x-\frac{3}{4}=\frac{3\sqrt{33}}{4} x-\frac{3}{4}=-\frac{3\sqrt{33}}{4}
Simplify.
x=\frac{3\sqrt{33}+3}{4} x=\frac{3-3\sqrt{33}}{4}
Add \frac{3}{4} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}