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x^{2}+3x=-\left(x^{2}-x\right)
Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by x-4, the least common multiple of x-4,4-x.
x^{2}+3x=-x^{2}+x
To find the opposite of x^{2}-x, find the opposite of each term.
x^{2}+3x+x^{2}=x
Add x^{2} to both sides.
2x^{2}+3x=x
Combine x^{2} and x^{2} to get 2x^{2}.
2x^{2}+3x-x=0
Subtract x from both sides.
2x^{2}+2x=0
Combine 3x and -x to get 2x.
x=\frac{-2±\sqrt{2^{2}}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±2}{2\times 2}
Take the square root of 2^{2}.
x=\frac{-2±2}{4}
Multiply 2 times 2.
x=\frac{0}{4}
Now solve the equation x=\frac{-2±2}{4} when ± is plus. Add -2 to 2.
x=0
Divide 0 by 4.
x=\frac{-4}{4}
Now solve the equation x=\frac{-2±2}{4} when ± is minus. Subtract 2 from -2.
x=-1
Divide -4 by 4.
x=0 x=-1
The equation is now solved.
x^{2}+3x=-\left(x^{2}-x\right)
Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by x-4, the least common multiple of x-4,4-x.
x^{2}+3x=-x^{2}+x
To find the opposite of x^{2}-x, find the opposite of each term.
x^{2}+3x+x^{2}=x
Add x^{2} to both sides.
2x^{2}+3x=x
Combine x^{2} and x^{2} to get 2x^{2}.
2x^{2}+3x-x=0
Subtract x from both sides.
2x^{2}+2x=0
Combine 3x and -x to get 2x.
\frac{2x^{2}+2x}{2}=\frac{0}{2}
Divide both sides by 2.
x^{2}+\frac{2}{2}x=\frac{0}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+x=\frac{0}{2}
Divide 2 by 2.
x^{2}+x=0
Divide 0 by 2.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+x+\frac{1}{4}=\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
\left(x+\frac{1}{2}\right)^{2}=\frac{1}{4}
Factor x^{2}+x+\frac{1}{4}. 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}{2}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
x+\frac{1}{2}=\frac{1}{2} x+\frac{1}{2}=-\frac{1}{2}
Simplify.
x=0 x=-1
Subtract \frac{1}{2} from both sides of the equation.