Solution Steps
Steps Using the Quadratic Formula
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Factor out 2.
a+b=-3 ab=1\left(-40\right)=-40
Consider x^{2}-3x-40. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-40. To find a and b, set up a system to be solved.
1,-40 2,-20 4,-10 5,-8
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 -40.
1-40=-39 2-20=-18 4-10=-6 5-8=-3
Calculate the sum for each pair.
a=-8 b=5
The solution is the pair that gives sum -3.
Rewrite x^{2}-3x-40 as \left(x^{2}-8x\right)+\left(5x-40\right).
Factor out x in the first and 5 in the second group.
Factor out common term x-8 by using distributive property.
Rewrite the complete factored expression.
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-80\right)}}{2\times 2}
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(-6\right)±\sqrt{36-4\times 2\left(-80\right)}}{2\times 2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36-8\left(-80\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-6\right)±\sqrt{36+640}}{2\times 2}
Multiply -8 times -80.
x=\frac{-\left(-6\right)±\sqrt{676}}{2\times 2}
Add 36 to 640.
x=\frac{-\left(-6\right)±26}{2\times 2}
Take the square root of 676.
x=\frac{6±26}{2\times 2}
The opposite of -6 is 6.
Multiply 2 times 2.
Now solve the equation x=\frac{6±26}{4} when ± is plus. Add 6 to 26.
Divide 32 by 4.
Now solve the equation x=\frac{6±26}{4} when ± is minus. Subtract 26 from 6.
Divide -20 by 4.
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -5 for x_{2}.
Simplify all the expressions of the form p-\left(-q\right) to p+q.