Solve for h
h=8
h=12
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h^{2}-20h+96=0
Add 96 to both sides.
a+b=-20 ab=96
To solve the equation, factor h^{2}-20h+96 using formula h^{2}+\left(a+b\right)h+ab=\left(h+a\right)\left(h+b\right). To find a and b, set up a system to be solved.
-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12
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 96.
-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20
Calculate the sum for each pair.
a=-12 b=-8
The solution is the pair that gives sum -20.
\left(h-12\right)\left(h-8\right)
Rewrite factored expression \left(h+a\right)\left(h+b\right) using the obtained values.
h=12 h=8
To find equation solutions, solve h-12=0 and h-8=0.
h^{2}-20h+96=0
Add 96 to both sides.
a+b=-20 ab=1\times 96=96
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as h^{2}+ah+bh+96. To find a and b, set up a system to be solved.
-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12
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 96.
-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20
Calculate the sum for each pair.
a=-12 b=-8
The solution is the pair that gives sum -20.
\left(h^{2}-12h\right)+\left(-8h+96\right)
Rewrite h^{2}-20h+96 as \left(h^{2}-12h\right)+\left(-8h+96\right).
h\left(h-12\right)-8\left(h-12\right)
Factor out h in the first and -8 in the second group.
\left(h-12\right)\left(h-8\right)
Factor out common term h-12 by using distributive property.
h=12 h=8
To find equation solutions, solve h-12=0 and h-8=0.
h^{2}-20h=-96
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.
h^{2}-20h-\left(-96\right)=-96-\left(-96\right)
Add 96 to both sides of the equation.
h^{2}-20h-\left(-96\right)=0
Subtracting -96 from itself leaves 0.
h^{2}-20h+96=0
Subtract -96 from 0.
h=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 96}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
h=\frac{-\left(-20\right)±\sqrt{400-4\times 96}}{2}
Square -20.
h=\frac{-\left(-20\right)±\sqrt{400-384}}{2}
Multiply -4 times 96.
h=\frac{-\left(-20\right)±\sqrt{16}}{2}
Add 400 to -384.
h=\frac{-\left(-20\right)±4}{2}
Take the square root of 16.
h=\frac{20±4}{2}
The opposite of -20 is 20.
h=\frac{24}{2}
Now solve the equation h=\frac{20±4}{2} when ± is plus. Add 20 to 4.
h=12
Divide 24 by 2.
h=\frac{16}{2}
Now solve the equation h=\frac{20±4}{2} when ± is minus. Subtract 4 from 20.
h=8
Divide 16 by 2.
h=12 h=8
The equation is now solved.
h^{2}-20h=-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.
h^{2}-20h+\left(-10\right)^{2}=-96+\left(-10\right)^{2}
Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
h^{2}-20h+100=-96+100
Square -10.
h^{2}-20h+100=4
Add -96 to 100.
\left(h-10\right)^{2}=4
Factor h^{2}-20h+100. 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(h-10\right)^{2}}=\sqrt{4}
Take the square root of both sides of the equation.
h-10=2 h-10=-2
Simplify.
h=12 h=8
Add 10 to both sides of the equation.
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Simultaneous equation
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Integration
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Limits
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