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x^{2}-10x-75=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-10 ab=-75
To solve the equation, factor x^{2}-10x-75 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,-75 3,-25 5,-15
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 -75.
1-75=-74 3-25=-22 5-15=-10
Calculate the sum for each pair.
a=-15 b=5
The solution is the pair that gives sum -10.
\left(x-15\right)\left(x+5\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=15 x=-5
To find equation solutions, solve x-15=0 and x+5=0.
x^{2}-10x-75=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-10 ab=1\left(-75\right)=-75
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-75. To find a and b, set up a system to be solved.
1,-75 3,-25 5,-15
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 -75.
1-75=-74 3-25=-22 5-15=-10
Calculate the sum for each pair.
a=-15 b=5
The solution is the pair that gives sum -10.
\left(x^{2}-15x\right)+\left(5x-75\right)
Rewrite x^{2}-10x-75 as \left(x^{2}-15x\right)+\left(5x-75\right).
x\left(x-15\right)+5\left(x-15\right)
Factor out x in the first and 5 in the second group.
\left(x-15\right)\left(x+5\right)
Factor out common term x-15 by using distributive property.
x=15 x=-5
To find equation solutions, solve x-15=0 and x+5=0.
x^{2}-10x-75=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(-75\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\left(-75\right)}}{2}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100+300}}{2}
Multiply -4 times -75.
x=\frac{-\left(-10\right)±\sqrt{400}}{2}
Add 100 to 300.
x=\frac{-\left(-10\right)±20}{2}
Take the square root of 400.
x=\frac{10±20}{2}
The opposite of -10 is 10.
x=\frac{30}{2}
Now solve the equation x=\frac{10±20}{2} when ± is plus. Add 10 to 20.
x=15
Divide 30 by 2.
x=-\frac{10}{2}
Now solve the equation x=\frac{10±20}{2} when ± is minus. Subtract 20 from 10.
x=-5
Divide -10 by 2.
x=15 x=-5
The equation is now solved.
x^{2}-10x-75=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-75-\left(-75\right)=-\left(-75\right)
Add 75 to both sides of the equation.
x^{2}-10x=-\left(-75\right)
Subtracting -75 from itself leaves 0.
x^{2}-10x=75
Subtract -75 from 0.
x^{2}-10x+\left(-5\right)^{2}=75+\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=75+25
Square -5.
x^{2}-10x+25=100
Add 75 to 25.
\left(x-5\right)^{2}=100
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{100}
Take the square root of both sides of the equation.
x-5=10 x-5=-10
Simplify.
x=15 x=-5
Add 5 to both sides of the equation.