Solve for x
x=-1
x=\frac{2}{9}\approx 0.222222222
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a+b=7 ab=9\left(-2\right)=-18
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-2. 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=-2 b=9
The solution is the pair that gives sum 7.
\left(9x^{2}-2x\right)+\left(9x-2\right)
Rewrite 9x^{2}+7x-2 as \left(9x^{2}-2x\right)+\left(9x-2\right).
x\left(9x-2\right)+9x-2
Factor out x in 9x^{2}-2x.
\left(9x-2\right)\left(x+1\right)
Factor out common term 9x-2 by using distributive property.
x=\frac{2}{9} x=-1
To find equation solutions, solve 9x-2=0 and x+1=0.
9x^{2}+7x-2=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{-7±\sqrt{7^{2}-4\times 9\left(-2\right)}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 7 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\times 9\left(-2\right)}}{2\times 9}
Square 7.
x=\frac{-7±\sqrt{49-36\left(-2\right)}}{2\times 9}
Multiply -4 times 9.
x=\frac{-7±\sqrt{49+72}}{2\times 9}
Multiply -36 times -2.
x=\frac{-7±\sqrt{121}}{2\times 9}
Add 49 to 72.
x=\frac{-7±11}{2\times 9}
Take the square root of 121.
x=\frac{-7±11}{18}
Multiply 2 times 9.
x=\frac{4}{18}
Now solve the equation x=\frac{-7±11}{18} when ± is plus. Add -7 to 11.
x=\frac{2}{9}
Reduce the fraction \frac{4}{18} to lowest terms by extracting and canceling out 2.
x=-\frac{18}{18}
Now solve the equation x=\frac{-7±11}{18} when ± is minus. Subtract 11 from -7.
x=-1
Divide -18 by 18.
x=\frac{2}{9} x=-1
The equation is now solved.
9x^{2}+7x-2=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.
9x^{2}+7x-2-\left(-2\right)=-\left(-2\right)
Add 2 to both sides of the equation.
9x^{2}+7x=-\left(-2\right)
Subtracting -2 from itself leaves 0.
9x^{2}+7x=2
Subtract -2 from 0.
\frac{9x^{2}+7x}{9}=\frac{2}{9}
Divide both sides by 9.
x^{2}+\frac{7}{9}x=\frac{2}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}+\frac{7}{9}x+\left(\frac{7}{18}\right)^{2}=\frac{2}{9}+\left(\frac{7}{18}\right)^{2}
Divide \frac{7}{9}, the coefficient of the x term, by 2 to get \frac{7}{18}. Then add the square of \frac{7}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{7}{9}x+\frac{49}{324}=\frac{2}{9}+\frac{49}{324}
Square \frac{7}{18} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{7}{9}x+\frac{49}{324}=\frac{121}{324}
Add \frac{2}{9} to \frac{49}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{7}{18}\right)^{2}=\frac{121}{324}
Factor x^{2}+\frac{7}{9}x+\frac{49}{324}. 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{7}{18}\right)^{2}}=\sqrt{\frac{121}{324}}
Take the square root of both sides of the equation.
x+\frac{7}{18}=\frac{11}{18} x+\frac{7}{18}=-\frac{11}{18}
Simplify.
x=\frac{2}{9} x=-1
Subtract \frac{7}{18} 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
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