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25\left(x^{2}-8x+15\right)
Factor out 25.
a+b=-8 ab=1\times 15=15
Consider x^{2}-8x+15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.
-1,-15 -3,-5
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 15.
-1-15=-16 -3-5=-8
Calculate the sum for each pair.
a=-5 b=-3
The solution is the pair that gives sum -8.
\left(x^{2}-5x\right)+\left(-3x+15\right)
Rewrite x^{2}-8x+15 as \left(x^{2}-5x\right)+\left(-3x+15\right).
x\left(x-5\right)-3\left(x-5\right)
Factor out x in the first and -3 in the second group.
\left(x-5\right)\left(x-3\right)
Factor out common term x-5 by using distributive property.
25\left(x-5\right)\left(x-3\right)
Rewrite the complete factored expression.
25x^{2}-200x+375=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{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\times 25\times 375}}{2\times 25}
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{-\left(-200\right)±\sqrt{40000-4\times 25\times 375}}{2\times 25}
Square -200.
x=\frac{-\left(-200\right)±\sqrt{40000-100\times 375}}{2\times 25}
Multiply -4 times 25.
x=\frac{-\left(-200\right)±\sqrt{40000-37500}}{2\times 25}
Multiply -100 times 375.
x=\frac{-\left(-200\right)±\sqrt{2500}}{2\times 25}
Add 40000 to -37500.
x=\frac{-\left(-200\right)±50}{2\times 25}
Take the square root of 2500.
x=\frac{200±50}{2\times 25}
The opposite of -200 is 200.
x=\frac{200±50}{50}
Multiply 2 times 25.
x=\frac{250}{50}
Now solve the equation x=\frac{200±50}{50} when ± is plus. Add 200 to 50.
x=5
Divide 250 by 50.
x=\frac{150}{50}
Now solve the equation x=\frac{200±50}{50} when ± is minus. Subtract 50 from 200.
x=3
Divide 150 by 50.
25x^{2}-200x+375=25\left(x-5\right)\left(x-3\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and 3 for x_{2}.
x ^ 2 -8x +15 = 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 25
r + s = 8 rs = 15
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) = 15
To solve for unknown quantity u, substitute these in the product equation rs = 15
16 - u^2 = 15
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 15-16 = -1
Simplify the expression by subtracting 16 on both sides
u^2 = 1 u = \pm\sqrt{1} = \pm 1
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =4 - 1 = 3 s = 4 + 1 = 5
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.