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a+b=15 ab=1\times 54=54
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+54. To find a and b, set up a system to be solved.
1,54 2,27 3,18 6,9
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 54.
1+54=55 2+27=29 3+18=21 6+9=15
Calculate the sum for each pair.
a=6 b=9
The solution is the pair that gives sum 15.
\left(x^{2}+6x\right)+\left(9x+54\right)
Rewrite x^{2}+15x+54 as \left(x^{2}+6x\right)+\left(9x+54\right).
x\left(x+6\right)+9\left(x+6\right)
Factor out x in the first and 9 in the second group.
\left(x+6\right)\left(x+9\right)
Factor out common term x+6 by using distributive property.
x^{2}+15x+54=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{-15±\sqrt{15^{2}-4\times 54}}{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{-15±\sqrt{225-4\times 54}}{2}
Square 15.
x=\frac{-15±\sqrt{225-216}}{2}
Multiply -4 times 54.
x=\frac{-15±\sqrt{9}}{2}
Add 225 to -216.
x=\frac{-15±3}{2}
Take the square root of 9.
x=-\frac{12}{2}
Now solve the equation x=\frac{-15±3}{2} when ± is plus. Add -15 to 3.
x=-6
Divide -12 by 2.
x=-\frac{18}{2}
Now solve the equation x=\frac{-15±3}{2} when ± is minus. Subtract 3 from -15.
x=-9
Divide -18 by 2.
x^{2}+15x+54=\left(x-\left(-6\right)\right)\left(x-\left(-9\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 -9 for x_{2}.
x^{2}+15x+54=\left(x+6\right)\left(x+9\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +15x +54 = 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 = -15 rs = 54
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{15}{2} - u s = -\frac{15}{2} + u
Two numbers r and s sum up to -15 exactly when the average of the two numbers is \frac{1}{2}*-15 = -\frac{15}{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{15}{2} - u) (-\frac{15}{2} + u) = 54
To solve for unknown quantity u, substitute these in the product equation rs = 54
\frac{225}{4} - u^2 = 54
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 54-\frac{225}{4} = -\frac{9}{4}
Simplify the expression by subtracting \frac{225}{4} on both sides
u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{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{15}{2} - \frac{3}{2} = -9 s = -\frac{15}{2} + \frac{3}{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.