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2x^{2}-x-3=0
Multiply both sides of the equation by 2.
a+b=-1 ab=2\left(-3\right)=-6
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-3. 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 negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.
1-6=-5 2-3=-1
Calculate the sum for each pair.
a=-3 b=2
The solution is the pair that gives sum -1.
\left(2x^{2}-3x\right)+\left(2x-3\right)
Rewrite 2x^{2}-x-3 as \left(2x^{2}-3x\right)+\left(2x-3\right).
x\left(2x-3\right)+2x-3
Factor out x in 2x^{2}-3x.
\left(2x-3\right)\left(x+1\right)
Factor out common term 2x-3 by using distributive property.
x=\frac{3}{2} x=-1
To find equation solutions, solve 2x-3=0 and x+1=0.
2x^{2}-x-3=0
Multiply both sides of the equation by 2.
x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-3\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1-8\left(-3\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-1\right)±\sqrt{1+24}}{2\times 2}
Multiply -8 times -3.
x=\frac{-\left(-1\right)±\sqrt{25}}{2\times 2}
Add 1 to 24.
x=\frac{-\left(-1\right)±5}{2\times 2}
Take the square root of 25.
x=\frac{1±5}{2\times 2}
The opposite of -1 is 1.
x=\frac{1±5}{4}
Multiply 2 times 2.
x=\frac{6}{4}
Now solve the equation x=\frac{1±5}{4} when ± is plus. Add 1 to 5.
x=\frac{3}{2}
Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.
x=-\frac{4}{4}
Now solve the equation x=\frac{1±5}{4} when ± is minus. Subtract 5 from 1.
x=-1
Divide -4 by 4.
x=\frac{3}{2} x=-1
The equation is now solved.
2x^{2}-x-3=0
Multiply both sides of the equation by 2.
2x^{2}-x=3
Add 3 to both sides. Anything plus zero gives itself.
\frac{2x^{2}-x}{2}=\frac{3}{2}
Divide both sides by 2.
x^{2}-\frac{1}{2}x=\frac{3}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{3}{2}+\left(-\frac{1}{4}\right)^{2}
Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{3}{2}+\frac{1}{16}
Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{25}{16}
Add \frac{3}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{4}\right)^{2}=\frac{25}{16}
Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}{4}\right)^{2}}=\sqrt{\frac{25}{16}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{5}{4} x-\frac{1}{4}=-\frac{5}{4}
Simplify.
x=\frac{3}{2} x=-1
Add \frac{1}{4} to both sides of the equation.