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-4x^{2}+6x-2=0
Subtract 2 from both sides.
-2x^{2}+3x-1=0
Divide both sides by 2.
a+b=3 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 positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.
\left(-2x^{2}+2x\right)+\left(x-1\right)
Rewrite -2x^{2}+3x-1 as \left(-2x^{2}+2x\right)+\left(x-1\right).
2x\left(-x+1\right)-\left(-x+1\right)
Factor out 2x in the first and -1 in the second group.
\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}+6x=2
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.
-4x^{2}+6x-2=2-2
Subtract 2 from both sides of the equation.
-4x^{2}+6x-2=0
Subtracting 2 from itself leaves 0.
x=\frac{-6±\sqrt{6^{2}-4\left(-4\right)\left(-2\right)}}{2\left(-4\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 6 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\left(-4\right)\left(-2\right)}}{2\left(-4\right)}
Square 6.
x=\frac{-6±\sqrt{36+16\left(-2\right)}}{2\left(-4\right)}
Multiply -4 times -4.
x=\frac{-6±\sqrt{36-32}}{2\left(-4\right)}
Multiply 16 times -2.
x=\frac{-6±\sqrt{4}}{2\left(-4\right)}
Add 36 to -32.
x=\frac{-6±2}{2\left(-4\right)}
Take the square root of 4.
x=\frac{-6±2}{-8}
Multiply 2 times -4.
x=-\frac{4}{-8}
Now solve the equation x=\frac{-6±2}{-8} when ± is plus. Add -6 to 2.
x=\frac{1}{2}
Reduce the fraction \frac{-4}{-8} to lowest terms by extracting and canceling out 4.
x=-\frac{8}{-8}
Now solve the equation x=\frac{-6±2}{-8} when ± is minus. Subtract 2 from -6.
x=1
Divide -8 by -8.
x=\frac{1}{2} x=1
The equation is now solved.
-4x^{2}+6x=2
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.
\frac{-4x^{2}+6x}{-4}=\frac{2}{-4}
Divide both sides by -4.
x^{2}+\frac{6}{-4}x=\frac{2}{-4}
Dividing by -4 undoes the multiplication by -4.
x^{2}-\frac{3}{2}x=\frac{2}{-4}
Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{3}{2}x=-\frac{1}{2}
Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{1}{2}+\left(-\frac{3}{4}\right)^{2}
Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{1}{2}+\frac{9}{16}
Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1}{16}
Add -\frac{1}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{3}{4}\right)^{2}=\frac{1}{16}
Factor x^{2}-\frac{3}{2}x+\frac{9}{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{3}{4}\right)^{2}}=\sqrt{\frac{1}{16}}
Take the square root of both sides of the equation.
x-\frac{3}{4}=\frac{1}{4} x-\frac{3}{4}=-\frac{1}{4}
Simplify.
x=1 x=\frac{1}{2}
Add \frac{3}{4} to both sides of the equation.