Solve for x
x=-3
x = \frac{3}{2} = 1\frac{1}{2} = 1.5
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a+b=3 ab=2\left(-9\right)=-18
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-9. To find a and b, set up a system to be solved.
-1,18 -2,9 -3,6
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -18.
-1+18=17 -2+9=7 -3+6=3
Calculate the sum for each pair.
a=-3 b=6
The solution is the pair that gives sum 3.
\left(2x^{2}-3x\right)+\left(6x-9\right)
Rewrite 2x^{2}+3x-9 as \left(2x^{2}-3x\right)+\left(6x-9\right).
x\left(2x-3\right)+3\left(2x-3\right)
Factor out x in the first and 3 in the second group.
\left(2x-3\right)\left(x+3\right)
Factor out common term 2x-3 by using distributive property.
x=\frac{3}{2} x=-3
To find equation solutions, solve 2x-3=0 and x+3=0.
2x^{2}+3x-9=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{-3±\sqrt{3^{2}-4\times 2\left(-9\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\times 2\left(-9\right)}}{2\times 2}
Square 3.
x=\frac{-3±\sqrt{9-8\left(-9\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-3±\sqrt{9+72}}{2\times 2}
Multiply -8 times -9.
x=\frac{-3±\sqrt{81}}{2\times 2}
Add 9 to 72.
x=\frac{-3±9}{2\times 2}
Take the square root of 81.
x=\frac{-3±9}{4}
Multiply 2 times 2.
x=\frac{6}{4}
Now solve the equation x=\frac{-3±9}{4} when ± is plus. Add -3 to 9.
x=\frac{3}{2}
Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.
x=-\frac{12}{4}
Now solve the equation x=\frac{-3±9}{4} when ± is minus. Subtract 9 from -3.
x=-3
Divide -12 by 4.
x=\frac{3}{2} x=-3
The equation is now solved.
2x^{2}+3x-9=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
2x^{2}+3x-9-\left(-9\right)=-\left(-9\right)
Add 9 to both sides of the equation.
2x^{2}+3x=-\left(-9\right)
Subtracting -9 from itself leaves 0.
2x^{2}+3x=9
Subtract -9 from 0.
\frac{2x^{2}+3x}{2}=\frac{9}{2}
Divide both sides by 2.
x^{2}+\frac{3}{2}x=\frac{9}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{9}{2}+\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}=\frac{9}{2}+\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{81}{16}
Add \frac{9}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{3}{4}\right)^{2}=\frac{81}{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{81}{16}}
Take the square root of both sides of the equation.
x+\frac{3}{4}=\frac{9}{4} x+\frac{3}{4}=-\frac{9}{4}
Simplify.
x=\frac{3}{2} x=-3
Subtract \frac{3}{4} from 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}