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3x^{2}-x=4
Subtract x from both sides.
3x^{2}-x-4=0
Subtract 4 from both sides.
a+b=-1 ab=3\left(-4\right)=-12
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-4. To find a and b, set up a system to be solved.
1,-12 2,-6 3,-4
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.
1-12=-11 2-6=-4 3-4=-1
Calculate the sum for each pair.
a=-4 b=3
The solution is the pair that gives sum -1.
\left(3x^{2}-4x\right)+\left(3x-4\right)
Rewrite 3x^{2}-x-4 as \left(3x^{2}-4x\right)+\left(3x-4\right).
x\left(3x-4\right)+3x-4
Factor out x in 3x^{2}-4x.
\left(3x-4\right)\left(x+1\right)
Factor out common term 3x-4 by using distributive property.
x=\frac{4}{3} x=-1
To find equation solutions, solve 3x-4=0 and x+1=0.
3x^{2}-x=4
Subtract x from both sides.
3x^{2}-x-4=0
Subtract 4 from both sides.
x=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-4\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1-12\left(-4\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times 3}
Multiply -12 times -4.
x=\frac{-\left(-1\right)±\sqrt{49}}{2\times 3}
Add 1 to 48.
x=\frac{-\left(-1\right)±7}{2\times 3}
Take the square root of 49.
x=\frac{1±7}{2\times 3}
The opposite of -1 is 1.
x=\frac{1±7}{6}
Multiply 2 times 3.
x=\frac{8}{6}
Now solve the equation x=\frac{1±7}{6} when ± is plus. Add 1 to 7.
x=\frac{4}{3}
Reduce the fraction \frac{8}{6} to lowest terms by extracting and canceling out 2.
x=-\frac{6}{6}
Now solve the equation x=\frac{1±7}{6} when ± is minus. Subtract 7 from 1.
x=-1
Divide -6 by 6.
x=\frac{4}{3} x=-1
The equation is now solved.
3x^{2}-x=4
Subtract x from both sides.
\frac{3x^{2}-x}{3}=\frac{4}{3}
Divide both sides by 3.
x^{2}-\frac{1}{3}x=\frac{4}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=\frac{4}{3}+\left(-\frac{1}{6}\right)^{2}
Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{4}{3}+\frac{1}{36}
Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{49}{36}
Add \frac{4}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{6}\right)^{2}=\frac{49}{36}
Factor x^{2}-\frac{1}{3}x+\frac{1}{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{1}{6}\right)^{2}}=\sqrt{\frac{49}{36}}
Take the square root of both sides of the equation.
x-\frac{1}{6}=\frac{7}{6} x-\frac{1}{6}=-\frac{7}{6}
Simplify.
x=\frac{4}{3} x=-1
Add \frac{1}{6} to both sides of the equation.