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x^{2}+4x=12
Multiply 9 and \frac{4}{3} to get 12.
x^{2}+4x-12=0
Subtract 12 from both sides.
a+b=4 ab=-12
To solve the equation, factor x^{2}+4x-12 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
-1,12 -2,6 -3,4
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. List all such integer pairs that give product -12.
-1+12=11 -2+6=4 -3+4=1
Calculate the sum for each pair.
a=-2 b=6
The solution is the pair that gives sum 4.
\left(x-2\right)\left(x+6\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=2 x=-6
To find equation solutions, solve x-2=0 and x+6=0.
x^{2}+4x=12
Multiply 9 and \frac{4}{3} to get 12.
x^{2}+4x-12=0
Subtract 12 from both sides.
a+b=4 ab=1\left(-12\right)=-12
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-12. To find a and b, set up a system to be solved.
-1,12 -2,6 -3,4
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. List all such integer pairs that give product -12.
-1+12=11 -2+6=4 -3+4=1
Calculate the sum for each pair.
a=-2 b=6
The solution is the pair that gives sum 4.
\left(x^{2}-2x\right)+\left(6x-12\right)
Rewrite x^{2}+4x-12 as \left(x^{2}-2x\right)+\left(6x-12\right).
x\left(x-2\right)+6\left(x-2\right)
Factor out x in the first and 6 in the second group.
\left(x-2\right)\left(x+6\right)
Factor out common term x-2 by using distributive property.
x=2 x=-6
To find equation solutions, solve x-2=0 and x+6=0.
x^{2}+4x=12
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^{2}+4x-12=12-12
Subtract 12 from both sides of the equation.
x^{2}+4x-12=0
Subtracting 12 from itself leaves 0.
x=\frac{-4±\sqrt{4^{2}-4\left(-12\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\left(-12\right)}}{2}
Square 4.
x=\frac{-4±\sqrt{16+48}}{2}
Multiply -4 times -12.
x=\frac{-4±\sqrt{64}}{2}
Add 16 to 48.
x=\frac{-4±8}{2}
Take the square root of 64.
x=\frac{4}{2}
Now solve the equation x=\frac{-4±8}{2} when ± is plus. Add -4 to 8.
x=2
Divide 4 by 2.
x=\frac{-12}{2}
Now solve the equation x=\frac{-4±8}{2} when ± is minus. Subtract 8 from -4.
x=-6
Divide -12 by 2.
x=2 x=-6
The equation is now solved.
x^{2}+4x=12
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.
x^{2}+4x+2^{2}=12+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=12+4
Square 2.
x^{2}+4x+4=16
Add 12 to 4.
\left(x+2\right)^{2}=16
Factor x^{2}+4x+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+2\right)^{2}}=\sqrt{16}
Take the square root of both sides of the equation.
x+2=4 x+2=-4
Simplify.
x=2 x=-6
Subtract 2 from both sides of the equation.