Solve for x
x=-2
x=\frac{1}{3}\approx 0.333333333
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3x^{2}+5x-4=-2
Swap sides so that all variable terms are on the left hand side.
3x^{2}+5x-4+2=0
Add 2 to both sides.
3x^{2}+5x-2=0
Add -4 and 2 to get -2.
a+b=5 ab=3\left(-2\right)=-6
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-2. To find a and b, set up a system to be solved.
-1,6 -2,3
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 -6.
-1+6=5 -2+3=1
Calculate the sum for each pair.
a=-1 b=6
The solution is the pair that gives sum 5.
\left(3x^{2}-x\right)+\left(6x-2\right)
Rewrite 3x^{2}+5x-2 as \left(3x^{2}-x\right)+\left(6x-2\right).
x\left(3x-1\right)+2\left(3x-1\right)
Factor out x in the first and 2 in the second group.
\left(3x-1\right)\left(x+2\right)
Factor out common term 3x-1 by using distributive property.
x=\frac{1}{3} x=-2
To find equation solutions, solve 3x-1=0 and x+2=0.
3x^{2}+5x-4=-2
Swap sides so that all variable terms are on the left hand side.
3x^{2}+5x-4+2=0
Add 2 to both sides.
3x^{2}+5x-2=0
Add -4 and 2 to get -2.
x=\frac{-5±\sqrt{5^{2}-4\times 3\left(-2\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 5 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-5±\sqrt{25-4\times 3\left(-2\right)}}{2\times 3}
Square 5.
x=\frac{-5±\sqrt{25-12\left(-2\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-5±\sqrt{25+24}}{2\times 3}
Multiply -12 times -2.
x=\frac{-5±\sqrt{49}}{2\times 3}
Add 25 to 24.
x=\frac{-5±7}{2\times 3}
Take the square root of 49.
x=\frac{-5±7}{6}
Multiply 2 times 3.
x=\frac{2}{6}
Now solve the equation x=\frac{-5±7}{6} when ± is plus. Add -5 to 7.
x=\frac{1}{3}
Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.
x=-\frac{12}{6}
Now solve the equation x=\frac{-5±7}{6} when ± is minus. Subtract 7 from -5.
x=-2
Divide -12 by 6.
x=\frac{1}{3} x=-2
The equation is now solved.
3x^{2}+5x-4=-2
Swap sides so that all variable terms are on the left hand side.
3x^{2}+5x=-2+4
Add 4 to both sides.
3x^{2}+5x=2
Add -2 and 4 to get 2.
\frac{3x^{2}+5x}{3}=\frac{2}{3}
Divide both sides by 3.
x^{2}+\frac{5}{3}x=\frac{2}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{2}{3}+\left(\frac{5}{6}\right)^{2}
Divide \frac{5}{3}, the coefficient of the x term, by 2 to get \frac{5}{6}. Then add the square of \frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}
Square \frac{5}{6} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}
Add \frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{5}{6}\right)^{2}=\frac{49}{36}
Factor x^{2}+\frac{5}{3}x+\frac{25}{36}. 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{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}
Take the square root of both sides of the equation.
x+\frac{5}{6}=\frac{7}{6} x+\frac{5}{6}=-\frac{7}{6}
Simplify.
x=\frac{1}{3} x=-2
Subtract \frac{5}{6} 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}