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a+b=-20 ab=64
To solve the equation, factor k^{2}-20k+64 using formula k^{2}+\left(a+b\right)k+ab=\left(k+a\right)\left(k+b\right). To find a and b, set up a system to be solved.
-1,-64 -2,-32 -4,-16 -8,-8
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 64.
-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16
Calculate the sum for each pair.
a=-16 b=-4
The solution is the pair that gives sum -20.
\left(k-16\right)\left(k-4\right)
Rewrite factored expression \left(k+a\right)\left(k+b\right) using the obtained values.
k=16 k=4
To find equation solutions, solve k-16=0 and k-4=0.
a+b=-20 ab=1\times 64=64
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as k^{2}+ak+bk+64. To find a and b, set up a system to be solved.
-1,-64 -2,-32 -4,-16 -8,-8
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 64.
-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16
Calculate the sum for each pair.
a=-16 b=-4
The solution is the pair that gives sum -20.
\left(k^{2}-16k\right)+\left(-4k+64\right)
Rewrite k^{2}-20k+64 as \left(k^{2}-16k\right)+\left(-4k+64\right).
k\left(k-16\right)-4\left(k-16\right)
Factor out k in the first and -4 in the second group.
\left(k-16\right)\left(k-4\right)
Factor out common term k-16 by using distributive property.
k=16 k=4
To find equation solutions, solve k-16=0 and k-4=0.
k^{2}-20k+64=0
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.
k=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 64}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
k=\frac{-\left(-20\right)±\sqrt{400-4\times 64}}{2}
Square -20.
k=\frac{-\left(-20\right)±\sqrt{400-256}}{2}
Multiply -4 times 64.
k=\frac{-\left(-20\right)±\sqrt{144}}{2}
Add 400 to -256.
k=\frac{-\left(-20\right)±12}{2}
Take the square root of 144.
k=\frac{20±12}{2}
The opposite of -20 is 20.
k=\frac{32}{2}
Now solve the equation k=\frac{20±12}{2} when ± is plus. Add 20 to 12.
k=16
Divide 32 by 2.
k=\frac{8}{2}
Now solve the equation k=\frac{20±12}{2} when ± is minus. Subtract 12 from 20.
k=4
Divide 8 by 2.
k=16 k=4
The equation is now solved.
k^{2}-20k+64=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
k^{2}-20k+64-64=-64
Subtract 64 from both sides of the equation.
k^{2}-20k=-64
Subtracting 64 from itself leaves 0.
k^{2}-20k+\left(-10\right)^{2}=-64+\left(-10\right)^{2}
Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
k^{2}-20k+100=-64+100
Square -10.
k^{2}-20k+100=36
Add -64 to 100.
\left(k-10\right)^{2}=36
Factor k^{2}-20k+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(k-10\right)^{2}}=\sqrt{36}
Take the square root of both sides of the equation.
k-10=6 k-10=-6
Simplify.
k=16 k=4
Add 10 to both sides of the equation.
x ^ 2 -20x +64 = 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 = 20 rs = 64
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 = 10 - u s = 10 + u
Two numbers r and s sum up to 20 exactly when the average of the two numbers is \frac{1}{2}*20 = 10. 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>
(10 - u) (10 + u) = 64
To solve for unknown quantity u, substitute these in the product equation rs = 64
100 - u^2 = 64
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 64-100 = -36
Simplify the expression by subtracting 100 on both sides
u^2 = 36 u = \pm\sqrt{36} = \pm 6
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =10 - 6 = 4 s = 10 + 6 = 16
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.