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2xx+x\left(-28\right)+96=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
2x^{2}+x\left(-28\right)+96=0
Multiply x and x to get x^{2}.
x^{2}-14x+48=0
Divide both sides by 2.
a+b=-14 ab=1\times 48=48
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+48. To find a and b, set up a system to be solved.
-1,-48 -2,-24 -3,-16 -4,-12 -6,-8
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 48.
-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14
Calculate the sum for each pair.
a=-8 b=-6
The solution is the pair that gives sum -14.
\left(x^{2}-8x\right)+\left(-6x+48\right)
Rewrite x^{2}-14x+48 as \left(x^{2}-8x\right)+\left(-6x+48\right).
x\left(x-8\right)-6\left(x-8\right)
Factor out x in the first and -6 in the second group.
\left(x-8\right)\left(x-6\right)
Factor out common term x-8 by using distributive property.
x=8 x=6
To find equation solutions, solve x-8=0 and x-6=0.
2xx+x\left(-28\right)+96=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
2x^{2}+x\left(-28\right)+96=0
Multiply x and x to get x^{2}.
2x^{2}-28x+96=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(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 96}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -28 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 96}}{2\times 2}
Square -28.
x=\frac{-\left(-28\right)±\sqrt{784-8\times 96}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-28\right)±\sqrt{784-768}}{2\times 2}
Multiply -8 times 96.
x=\frac{-\left(-28\right)±\sqrt{16}}{2\times 2}
Add 784 to -768.
x=\frac{-\left(-28\right)±4}{2\times 2}
Take the square root of 16.
x=\frac{28±4}{2\times 2}
The opposite of -28 is 28.
x=\frac{28±4}{4}
Multiply 2 times 2.
x=\frac{32}{4}
Now solve the equation x=\frac{28±4}{4} when ± is plus. Add 28 to 4.
x=8
Divide 32 by 4.
x=\frac{24}{4}
Now solve the equation x=\frac{28±4}{4} when ± is minus. Subtract 4 from 28.
x=6
Divide 24 by 4.
x=8 x=6
The equation is now solved.
2xx+x\left(-28\right)+96=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
2x^{2}+x\left(-28\right)+96=0
Multiply x and x to get x^{2}.
2x^{2}+x\left(-28\right)=-96
Subtract 96 from both sides. Anything subtracted from zero gives its negation.
2x^{2}-28x=-96
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{2x^{2}-28x}{2}=-\frac{96}{2}
Divide both sides by 2.
x^{2}+\left(-\frac{28}{2}\right)x=-\frac{96}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-14x=-\frac{96}{2}
Divide -28 by 2.
x^{2}-14x=-48
Divide -96 by 2.
x^{2}-14x+\left(-7\right)^{2}=-48+\left(-7\right)^{2}
Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-14x+49=-48+49
Square -7.
x^{2}-14x+49=1
Add -48 to 49.
\left(x-7\right)^{2}=1
Factor x^{2}-14x+49. 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-7\right)^{2}}=\sqrt{1}
Take the square root of both sides of the equation.
x-7=1 x-7=-1
Simplify.
x=8 x=6
Add 7 to both sides of the equation.