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5\left(-x^{2}+8x-16\right)
Factor out 5.
a+b=8 ab=-\left(-16\right)=16
Consider -x^{2}+8x-16. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-16. To find a and b, set up a system to be solved.
1,16 2,8 4,4
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 16.
1+16=17 2+8=10 4+4=8
Calculate the sum for each pair.
a=4 b=4
The solution is the pair that gives sum 8.
\left(-x^{2}+4x\right)+\left(4x-16\right)
Rewrite -x^{2}+8x-16 as \left(-x^{2}+4x\right)+\left(4x-16\right).
-x\left(x-4\right)+4\left(x-4\right)
Factor out -x in the first and 4 in the second group.
\left(x-4\right)\left(-x+4\right)
Factor out common term x-4 by using distributive property.
5\left(x-4\right)\left(-x+4\right)
Rewrite the complete factored expression.
-5x^{2}+40x-80=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{-40±\sqrt{40^{2}-4\left(-5\right)\left(-80\right)}}{2\left(-5\right)}
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{-40±\sqrt{1600-4\left(-5\right)\left(-80\right)}}{2\left(-5\right)}
Square 40.
x=\frac{-40±\sqrt{1600+20\left(-80\right)}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-40±\sqrt{1600-1600}}{2\left(-5\right)}
Multiply 20 times -80.
x=\frac{-40±\sqrt{0}}{2\left(-5\right)}
Add 1600 to -1600.
x=\frac{-40±0}{2\left(-5\right)}
Take the square root of 0.
x=\frac{-40±0}{-10}
Multiply 2 times -5.
-5x^{2}+40x-80=-5\left(x-4\right)\left(x-4\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and 4 for x_{2}.
x ^ 2 -8x +16 = 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 = 8 rs = 16
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 = 4 - u s = 4 + u
Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 4. 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>
(4 - u) (4 + u) = 16
To solve for unknown quantity u, substitute these in the product equation rs = 16
16 - u^2 = 16
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 16-16 = 0
Simplify the expression by subtracting 16 on both sides
u^2 = 0 u = 0
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r = s = 4
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.