Solution Steps
Steps Using the Quadratic Formula
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Factor out 3.
a+b=-11 ab=1\times 24=24
Consider x^{2}-11x+24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+24. To find a and b, set up a system to be solved.
-1,-24 -2,-12 -3,-8 -4,-6
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 24.
-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10
Calculate the sum for each pair.
a=-8 b=-3
The solution is the pair that gives sum -11.
Rewrite x^{2}-11x+24 as \left(x^{2}-8x\right)+\left(-3x+24\right).
Factor out x in the first and -3 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(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 3\times 72}}{2\times 3}
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(-33\right)±\sqrt{1089-4\times 3\times 72}}{2\times 3}
Square -33.
x=\frac{-\left(-33\right)±\sqrt{1089-12\times 72}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-33\right)±\sqrt{1089-864}}{2\times 3}
Multiply -12 times 72.
x=\frac{-\left(-33\right)±\sqrt{225}}{2\times 3}
Add 1089 to -864.
x=\frac{-\left(-33\right)±15}{2\times 3}
Take the square root of 225.
x=\frac{33±15}{2\times 3}
The opposite of -33 is 33.
Multiply 2 times 3.
Now solve the equation x=\frac{33±15}{6} when ± is plus. Add 33 to 15.
Divide 48 by 6.
Now solve the equation x=\frac{33±15}{6} when ± is minus. Subtract 15 from 33.
Divide 18 by 6.
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 3 for x_{2}.