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2\left(x^{2}-10x+21\right)
Factor out 2.
a+b=-10 ab=1\times 21=21
Consider x^{2}-10x+21. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+21. To find a and b, set up a system to be solved.
-1,-21 -3,-7
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 21.
-1-21=-22 -3-7=-10
Calculate the sum for each pair.
a=-7 b=-3
The solution is the pair that gives sum -10.
\left(x^{2}-7x\right)+\left(-3x+21\right)
Rewrite x^{2}-10x+21 as \left(x^{2}-7x\right)+\left(-3x+21\right).
x\left(x-7\right)-3\left(x-7\right)
Factor out x in the first and -3 in the second group.
\left(x-7\right)\left(x-3\right)
Factor out common term x-7 by using distributive property.
2\left(x-7\right)\left(x-3\right)
Rewrite the complete factored expression.
2x^{2}-20x+42=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(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 2\times 42}}{2\times 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{-\left(-20\right)±\sqrt{400-4\times 2\times 42}}{2\times 2}
Square -20.
x=\frac{-\left(-20\right)±\sqrt{400-8\times 42}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-20\right)±\sqrt{400-336}}{2\times 2}
Multiply -8 times 42.
x=\frac{-\left(-20\right)±\sqrt{64}}{2\times 2}
Add 400 to -336.
x=\frac{-\left(-20\right)±8}{2\times 2}
Take the square root of 64.
x=\frac{20±8}{2\times 2}
The opposite of -20 is 20.
x=\frac{20±8}{4}
Multiply 2 times 2.
x=\frac{28}{4}
Now solve the equation x=\frac{20±8}{4} when ± is plus. Add 20 to 8.
x=7
Divide 28 by 4.
x=\frac{12}{4}
Now solve the equation x=\frac{20±8}{4} when ± is minus. Subtract 8 from 20.
x=3
Divide 12 by 4.
2x^{2}-20x+42=2\left(x-7\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 7 for x_{1} and 3 for x_{2}.
x ^ 2 -10x +21 = 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 2
r + s = 10 rs = 21
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 = 5 - u s = 5 + u
Two numbers r and s sum up to 10 exactly when the average of the two numbers is \frac{1}{2}*10 = 5. 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>
(5 - u) (5 + u) = 21
To solve for unknown quantity u, substitute these in the product equation rs = 21
25 - u^2 = 21
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 21-25 = -4
Simplify the expression by subtracting 25 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 =5 - 2 = 3 s = 5 + 2 = 7
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.