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a+b=16 ab=1\times 60=60
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+60. To find a and b, set up a system to be solved.
1,60 2,30 3,20 4,15 5,12 6,10
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 60.
1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16
Calculate the sum for each pair.
a=6 b=10
The solution is the pair that gives sum 16.
\left(x^{2}+6x\right)+\left(10x+60\right)
Rewrite x^{2}+16x+60 as \left(x^{2}+6x\right)+\left(10x+60\right).
x\left(x+6\right)+10\left(x+6\right)
Factor out x in the first and 10 in the second group.
\left(x+6\right)\left(x+10\right)
Factor out common term x+6 by using distributive property.
x^{2}+16x+60=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{-16±\sqrt{16^{2}-4\times 60}}{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{-16±\sqrt{256-4\times 60}}{2}
Square 16.
x=\frac{-16±\sqrt{256-240}}{2}
Multiply -4 times 60.
x=\frac{-16±\sqrt{16}}{2}
Add 256 to -240.
x=\frac{-16±4}{2}
Take the square root of 16.
x=-\frac{12}{2}
Now solve the equation x=\frac{-16±4}{2} when ± is plus. Add -16 to 4.
x=-6
Divide -12 by 2.
x=-\frac{20}{2}
Now solve the equation x=\frac{-16±4}{2} when ± is minus. Subtract 4 from -16.
x=-10
Divide -20 by 2.
x^{2}+16x+60=\left(x-\left(-6\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 -6 for x_{1} and -10 for x_{2}.
x^{2}+16x+60=\left(x+6\right)\left(x+10\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +16x +60 = 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.
r + s = -16 rs = 60
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 = -8 - u s = -8 + u
Two numbers r and s sum up to -16 exactly when the average of the two numbers is \frac{1}{2}*-16 = -8. 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>
(-8 - u) (-8 + u) = 60
To solve for unknown quantity u, substitute these in the product equation rs = 60
64 - u^2 = 60
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 60-64 = -4
Simplify the expression by subtracting 64 on both sides
u^2 = 4 u = \pm\sqrt{4} = \pm 2
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-8 - 2 = -10 s = -8 + 2 = -6
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.