Factor
\left(g-13\right)\left(g-3\right)
Evaluate
\left(g-13\right)\left(g-3\right)
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a+b=-16 ab=1\times 39=39
Factor the expression by grouping. First, the expression needs to be rewritten as g^{2}+ag+bg+39. To find a and b, set up a system to be solved.
-1,-39 -3,-13
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 39.
-1-39=-40 -3-13=-16
Calculate the sum for each pair.
a=-13 b=-3
The solution is the pair that gives sum -16.
\left(g^{2}-13g\right)+\left(-3g+39\right)
Rewrite g^{2}-16g+39 as \left(g^{2}-13g\right)+\left(-3g+39\right).
g\left(g-13\right)-3\left(g-13\right)
Factor out g in the first and -3 in the second group.
\left(g-13\right)\left(g-3\right)
Factor out common term g-13 by using distributive property.
g^{2}-16g+39=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.
g=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 39}}{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.
g=\frac{-\left(-16\right)±\sqrt{256-4\times 39}}{2}
Square -16.
g=\frac{-\left(-16\right)±\sqrt{256-156}}{2}
Multiply -4 times 39.
g=\frac{-\left(-16\right)±\sqrt{100}}{2}
Add 256 to -156.
g=\frac{-\left(-16\right)±10}{2}
Take the square root of 100.
g=\frac{16±10}{2}
The opposite of -16 is 16.
g=\frac{26}{2}
Now solve the equation g=\frac{16±10}{2} when ± is plus. Add 16 to 10.
g=13
Divide 26 by 2.
g=\frac{6}{2}
Now solve the equation g=\frac{16±10}{2} when ± is minus. Subtract 10 from 16.
g=3
Divide 6 by 2.
g^{2}-16g+39=\left(g-13\right)\left(g-3\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13 for x_{1} and 3 for x_{2}.
x ^ 2 -16x +39 = 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 = 39
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) = 39
To solve for unknown quantity u, substitute these in the product equation rs = 39
64 - u^2 = 39
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 39-64 = -25
Simplify the expression by subtracting 64 on both sides
u^2 = 25 u = \pm\sqrt{25} = \pm 5
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =8 - 5 = 3 s = 8 + 5 = 13
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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