Solve for x
x=-20
x=30
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a+b=-10 ab=-600
To solve the equation, factor x^{2}-10x-600 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,-600 2,-300 3,-200 4,-150 5,-120 6,-100 8,-75 10,-60 12,-50 15,-40 20,-30 24,-25
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -600.
1-600=-599 2-300=-298 3-200=-197 4-150=-146 5-120=-115 6-100=-94 8-75=-67 10-60=-50 12-50=-38 15-40=-25 20-30=-10 24-25=-1
Calculate the sum for each pair.
a=-30 b=20
The solution is the pair that gives sum -10.
\left(x-30\right)\left(x+20\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=30 x=-20
To find equation solutions, solve x-30=0 and x+20=0.
a+b=-10 ab=1\left(-600\right)=-600
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-600. To find a and b, set up a system to be solved.
1,-600 2,-300 3,-200 4,-150 5,-120 6,-100 8,-75 10,-60 12,-50 15,-40 20,-30 24,-25
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -600.
1-600=-599 2-300=-298 3-200=-197 4-150=-146 5-120=-115 6-100=-94 8-75=-67 10-60=-50 12-50=-38 15-40=-25 20-30=-10 24-25=-1
Calculate the sum for each pair.
a=-30 b=20
The solution is the pair that gives sum -10.
\left(x^{2}-30x\right)+\left(20x-600\right)
Rewrite x^{2}-10x-600 as \left(x^{2}-30x\right)+\left(20x-600\right).
x\left(x-30\right)+20\left(x-30\right)
Factor out x in the first and 20 in the second group.
\left(x-30\right)\left(x+20\right)
Factor out common term x-30 by using distributive property.
x=30 x=-20
To find equation solutions, solve x-30=0 and x+20=0.
x^{2}-10x-600=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(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-600\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\left(-600\right)}}{2}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100+2400}}{2}
Multiply -4 times -600.
x=\frac{-\left(-10\right)±\sqrt{2500}}{2}
Add 100 to 2400.
x=\frac{-\left(-10\right)±50}{2}
Take the square root of 2500.
x=\frac{10±50}{2}
The opposite of -10 is 10.
x=\frac{60}{2}
Now solve the equation x=\frac{10±50}{2} when ± is plus. Add 10 to 50.
x=30
Divide 60 by 2.
x=-\frac{40}{2}
Now solve the equation x=\frac{10±50}{2} when ± is minus. Subtract 50 from 10.
x=-20
Divide -40 by 2.
x=30 x=-20
The equation is now solved.
x^{2}-10x-600=0
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}-10x-600-\left(-600\right)=-\left(-600\right)
Add 600 to both sides of the equation.
x^{2}-10x=-\left(-600\right)
Subtracting -600 from itself leaves 0.
x^{2}-10x=600
Subtract -600 from 0.
x^{2}-10x+\left(-5\right)^{2}=600+\left(-5\right)^{2}
Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-10x+25=600+25
Square -5.
x^{2}-10x+25=625
Add 600 to 25.
\left(x-5\right)^{2}=625
Factor x^{2}-10x+25. 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-5\right)^{2}}=\sqrt{625}
Take the square root of both sides of the equation.
x-5=25 x-5=-25
Simplify.
x=30 x=-20
Add 5 to both sides of the equation.
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