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2x^{2}+1=x^{2}-3x
Use the distributive property to multiply x by x-3.
2x^{2}+1-x^{2}=-3x
Subtract x^{2} from both sides.
x^{2}+1=-3x
Combine 2x^{2} and -x^{2} to get x^{2}.
x^{2}+1+3x=0
Add 3x to both sides.
x^{2}+3x+1=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{-3±\sqrt{3^{2}-4}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4}}{2}
Square 3.
x=\frac{-3±\sqrt{5}}{2}
Add 9 to -4.
x=\frac{\sqrt{5}-3}{2}
Now solve the equation x=\frac{-3±\sqrt{5}}{2} when ± is plus. Add -3 to \sqrt{5}.
x=\frac{-\sqrt{5}-3}{2}
Now solve the equation x=\frac{-3±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from -3.
x=\frac{\sqrt{5}-3}{2} x=\frac{-\sqrt{5}-3}{2}
The equation is now solved.
2x^{2}+1=x^{2}-3x
Use the distributive property to multiply x by x-3.
2x^{2}+1-x^{2}=-3x
Subtract x^{2} from both sides.
x^{2}+1=-3x
Combine 2x^{2} and -x^{2} to get x^{2}.
x^{2}+1+3x=0
Add 3x to both sides.
x^{2}+3x=-1
Subtract 1 from both sides. Anything subtracted from zero gives its negation.
x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-1+\left(\frac{3}{2}\right)^{2}
Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+3x+\frac{9}{4}=-1+\frac{9}{4}
Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+3x+\frac{9}{4}=\frac{5}{4}
Add -1 to \frac{9}{4}.
\left(x+\frac{3}{2}\right)^{2}=\frac{5}{4}
Factor x^{2}+3x+\frac{9}{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{3}{2}\right)^{2}}=\sqrt{\frac{5}{4}}
Take the square root of both sides of the equation.
x+\frac{3}{2}=\frac{\sqrt{5}}{2} x+\frac{3}{2}=-\frac{\sqrt{5}}{2}
Simplify.
x=\frac{\sqrt{5}-3}{2} x=\frac{-\sqrt{5}-3}{2}
Subtract \frac{3}{2} from both sides of the equation.