Factor
3\left(x+1\right)\left(x+10\right)
Evaluate
3\left(x+1\right)\left(x+10\right)
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3\left(x^{2}+11x+10\right)
Factor out 3.
a+b=11 ab=1\times 10=10
Consider x^{2}+11x+10. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+10. To find a and b, set up a system to be solved.
1,10 2,5
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 10.
1+10=11 2+5=7
Calculate the sum for each pair.
a=1 b=10
The solution is the pair that gives sum 11.
\left(x^{2}+x\right)+\left(10x+10\right)
Rewrite x^{2}+11x+10 as \left(x^{2}+x\right)+\left(10x+10\right).
x\left(x+1\right)+10\left(x+1\right)
Factor out x in the first and 10 in the second group.
\left(x+1\right)\left(x+10\right)
Factor out common term x+1 by using distributive property.
3\left(x+1\right)\left(x+10\right)
Rewrite the complete factored expression.
3x^{2}+33x+30=0
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{-33±\sqrt{33^{2}-4\times 3\times 30}}{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{-33±\sqrt{1089-4\times 3\times 30}}{2\times 3}
Square 33.
x=\frac{-33±\sqrt{1089-12\times 30}}{2\times 3}
Multiply -4 times 3.
x=\frac{-33±\sqrt{1089-360}}{2\times 3}
Multiply -12 times 30.
x=\frac{-33±\sqrt{729}}{2\times 3}
Add 1089 to -360.
x=\frac{-33±27}{2\times 3}
Take the square root of 729.
x=\frac{-33±27}{6}
Multiply 2 times 3.
x=-\frac{6}{6}
Now solve the equation x=\frac{-33±27}{6} when ± is plus. Add -33 to 27.
x=-1
Divide -6 by 6.
x=-\frac{60}{6}
Now solve the equation x=\frac{-33±27}{6} when ± is minus. Subtract 27 from -33.
x=-10
Divide -60 by 6.
3x^{2}+33x+30=3\left(x-\left(-1\right)\right)\left(x-\left(-10\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -10 for x_{2}.
3x^{2}+33x+30=3\left(x+1\right)\left(x+10\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +11x +10 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 3
r + s = -11 rs = 10
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{11}{2} - u s = -\frac{11}{2} + u
Two numbers r and s sum up to -11 exactly when the average of the two numbers is \frac{1}{2}*-11 = -\frac{11}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{11}{2} - u) (-\frac{11}{2} + u) = 10
To solve for unknown quantity u, substitute these in the product equation rs = 10
\frac{121}{4} - u^2 = 10
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 10-\frac{121}{4} = -\frac{81}{4}
Simplify the expression by subtracting \frac{121}{4} on both sides
u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{11}{2} - \frac{9}{2} = -10 s = -\frac{11}{2} + \frac{9}{2} = -1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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