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3x^{2}+5x-2=0
Subtract 2 from both sides.
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=2
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.
3x^{2}+5x-2=2-2
Subtract 2 from both sides of the equation.
3x^{2}+5x-2=0
Subtracting 2 from itself leaves 0.
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=2
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.
\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.