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x\times 3+x^{2}-88=0
Subtract 88 from both sides.
x^{2}+3x-88=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=3 ab=-88
To solve the equation, factor x^{2}+3x-88 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.
-1,88 -2,44 -4,22 -8,11
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 -88.
-1+88=87 -2+44=42 -4+22=18 -8+11=3
Calculate the sum for each pair.
a=-8 b=11
The solution is the pair that gives sum 3.
\left(x-8\right)\left(x+11\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=8 x=-11
To find equation solutions, solve x-8=0 and x+11=0.
x\times 3+x^{2}-88=0
Subtract 88 from both sides.
x^{2}+3x-88=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=3 ab=1\left(-88\right)=-88
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-88. To find a and b, set up a system to be solved.
-1,88 -2,44 -4,22 -8,11
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 -88.
-1+88=87 -2+44=42 -4+22=18 -8+11=3
Calculate the sum for each pair.
a=-8 b=11
The solution is the pair that gives sum 3.
\left(x^{2}-8x\right)+\left(11x-88\right)
Rewrite x^{2}+3x-88 as \left(x^{2}-8x\right)+\left(11x-88\right).
x\left(x-8\right)+11\left(x-8\right)
Factor out x in the first and 11 in the second group.
\left(x-8\right)\left(x+11\right)
Factor out common term x-8 by using distributive property.
x=8 x=-11
To find equation solutions, solve x-8=0 and x+11=0.
x^{2}+3x=88
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}+3x-88=88-88
Subtract 88 from both sides of the equation.
x^{2}+3x-88=0
Subtracting 88 from itself leaves 0.
x=\frac{-3±\sqrt{3^{2}-4\left(-88\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -88 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-88\right)}}{2}
Square 3.
x=\frac{-3±\sqrt{9+352}}{2}
Multiply -4 times -88.
x=\frac{-3±\sqrt{361}}{2}
Add 9 to 352.
x=\frac{-3±19}{2}
Take the square root of 361.
x=\frac{16}{2}
Now solve the equation x=\frac{-3±19}{2} when ± is plus. Add -3 to 19.
x=8
Divide 16 by 2.
x=-\frac{22}{2}
Now solve the equation x=\frac{-3±19}{2} when ± is minus. Subtract 19 from -3.
x=-11
Divide -22 by 2.
x=8 x=-11
The equation is now solved.
x^{2}+3x=88
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}+3x+\left(\frac{3}{2}\right)^{2}=88+\left(\frac{3}{2}\right)^{2}
Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+3x+\frac{9}{4}=88+\frac{9}{4}
Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+3x+\frac{9}{4}=\frac{361}{4}
Add 88 to \frac{9}{4}.
\left(x+\frac{3}{2}\right)^{2}=\frac{361}{4}
Factor x^{2}+3x+\frac{9}{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+\frac{3}{2}\right)^{2}}=\sqrt{\frac{361}{4}}
Take the square root of both sides of the equation.
x+\frac{3}{2}=\frac{19}{2} x+\frac{3}{2}=-\frac{19}{2}
Simplify.
x=8 x=-11
Subtract \frac{3}{2} from both sides of the equation.