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2x^{2}-x-1=0
Divide both sides by 2.
a+b=-1 ab=2\left(-1\right)=-2
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-1. To find a and b, set up a system to be solved.
a=-2 b=1
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. The only such pair is the system solution.
\left(2x^{2}-2x\right)+\left(x-1\right)
Rewrite 2x^{2}-x-1 as \left(2x^{2}-2x\right)+\left(x-1\right).
2x\left(x-1\right)+x-1
Factor out 2x in 2x^{2}-2x.
\left(x-1\right)\left(2x+1\right)
Factor out common term x-1 by using distributive property.
x=1 x=-\frac{1}{2}
To find equation solutions, solve x-1=0 and 2x+1=0.
4x^{2}-2x-2=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{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\left(-2\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -2 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4\times 4\left(-2\right)}}{2\times 4}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{4-16\left(-2\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\times 4}
Multiply -16 times -2.
x=\frac{-\left(-2\right)±\sqrt{36}}{2\times 4}
Add 4 to 32.
x=\frac{-\left(-2\right)±6}{2\times 4}
Take the square root of 36.
x=\frac{2±6}{2\times 4}
The opposite of -2 is 2.
x=\frac{2±6}{8}
Multiply 2 times 4.
x=\frac{8}{8}
Now solve the equation x=\frac{2±6}{8} when ± is plus. Add 2 to 6.
x=1
Divide 8 by 8.
x=-\frac{4}{8}
Now solve the equation x=\frac{2±6}{8} when ± is minus. Subtract 6 from 2.
x=-\frac{1}{2}
Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.
x=1 x=-\frac{1}{2}
The equation is now solved.
4x^{2}-2x-2=0
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.
4x^{2}-2x-2-\left(-2\right)=-\left(-2\right)
Add 2 to both sides of the equation.
4x^{2}-2x=-\left(-2\right)
Subtracting -2 from itself leaves 0.
4x^{2}-2x=2
Subtract -2 from 0.
\frac{4x^{2}-2x}{4}=\frac{2}{4}
Divide both sides by 4.
x^{2}+\left(-\frac{2}{4}\right)x=\frac{2}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{1}{2}x=\frac{2}{4}
Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{1}{2}x=\frac{1}{2}
Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{1}{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{1}{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{9}{16}
Add \frac{1}{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{9}{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{9}{16}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{3}{4} x-\frac{1}{4}=-\frac{3}{4}
Simplify.
x=1 x=-\frac{1}{2}
Add \frac{1}{4} to both sides of the equation.