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-4x^{2}+8x+12=0
Multiply both sides of the equation by 3.
-x^{2}+2x+3=0
Divide both sides by 4.
a+b=2 ab=-3=-3
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+3. To find a and b, set up a system to be solved.
a=3 b=-1
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.
\left(-x^{2}+3x\right)+\left(-x+3\right)
Rewrite -x^{2}+2x+3 as \left(-x^{2}+3x\right)+\left(-x+3\right).
-x\left(x-3\right)-\left(x-3\right)
Factor out -x in the first and -1 in the second group.
\left(x-3\right)\left(-x-1\right)
Factor out common term x-3 by using distributive property.
x=3 x=-1
To find equation solutions, solve x-3=0 and -x-1=0.
-4x^{2}+8x+12=0
Multiply both sides of the equation by 3.
x=\frac{-8±\sqrt{8^{2}-4\left(-4\right)\times 12}}{2\left(-4\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 8 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\left(-4\right)\times 12}}{2\left(-4\right)}
Square 8.
x=\frac{-8±\sqrt{64+16\times 12}}{2\left(-4\right)}
Multiply -4 times -4.
x=\frac{-8±\sqrt{64+192}}{2\left(-4\right)}
Multiply 16 times 12.
x=\frac{-8±\sqrt{256}}{2\left(-4\right)}
Add 64 to 192.
x=\frac{-8±16}{2\left(-4\right)}
Take the square root of 256.
x=\frac{-8±16}{-8}
Multiply 2 times -4.
x=\frac{8}{-8}
Now solve the equation x=\frac{-8±16}{-8} when ± is plus. Add -8 to 16.
x=-1
Divide 8 by -8.
x=-\frac{24}{-8}
Now solve the equation x=\frac{-8±16}{-8} when ± is minus. Subtract 16 from -8.
x=3
Divide -24 by -8.
x=-1 x=3
The equation is now solved.
-4x^{2}+8x+12=0
Multiply both sides of the equation by 3.
-4x^{2}+8x=-12
Subtract 12 from both sides. Anything subtracted from zero gives its negation.
\frac{-4x^{2}+8x}{-4}=-\frac{12}{-4}
Divide both sides by -4.
x^{2}+\frac{8}{-4}x=-\frac{12}{-4}
Dividing by -4 undoes the multiplication by -4.
x^{2}-2x=-\frac{12}{-4}
Divide 8 by -4.
x^{2}-2x=3
Divide -12 by -4.
x^{2}-2x+1=3+1
Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-2x+1=4
Add 3 to 1.
\left(x-1\right)^{2}=4
Factor x^{2}-2x+1. 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-1\right)^{2}}=\sqrt{4}
Take the square root of both sides of the equation.
x-1=2 x-1=-2
Simplify.
x=3 x=-1
Add 1 to both sides of the equation.