Solve for x
x=4
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x^{2}-8x+16=0
Divide both sides by 2.
a+b=-8 ab=1\times 16=16
To solve the equation, factor the left hand side by grouping. First, left hand side 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 negative, a and b are both negative. 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.
\left(x-4\right)^{2}
Rewrite as a binomial square.
x=4
To find equation solution, solve x-4=0.
2x^{2}-16x+32=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.
x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 2\times 32}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -16 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-16\right)±\sqrt{256-4\times 2\times 32}}{2\times 2}
Square -16.
x=\frac{-\left(-16\right)±\sqrt{256-8\times 32}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-16\right)±\sqrt{256-256}}{2\times 2}
Multiply -8 times 32.
x=\frac{-\left(-16\right)±\sqrt{0}}{2\times 2}
Add 256 to -256.
x=-\frac{-16}{2\times 2}
Take the square root of 0.
x=\frac{16}{2\times 2}
The opposite of -16 is 16.
x=\frac{16}{4}
Multiply 2 times 2.
x=4
Divide 16 by 4.
2x^{2}-16x+32=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.
2x^{2}-16x+32-32=-32
Subtract 32 from both sides of the equation.
2x^{2}-16x=-32
Subtracting 32 from itself leaves 0.
\frac{2x^{2}-16x}{2}=-\frac{32}{2}
Divide both sides by 2.
x^{2}+\left(-\frac{16}{2}\right)x=-\frac{32}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-8x=-\frac{32}{2}
Divide -16 by 2.
x^{2}-8x=-16
Divide -32 by 2.
x^{2}-8x+\left(-4\right)^{2}=-16+\left(-4\right)^{2}
Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-8x+16=-16+16
Square -4.
x^{2}-8x+16=0
Add -16 to 16.
\left(x-4\right)^{2}=0
Factor x^{2}-8x+16. 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(x-4\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-4=0 x-4=0
Simplify.
x=4 x=4
Add 4 to both sides of the equation.
x=4
The equation is now solved. Solutions are the same.
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.This is achieved by dividing both sides of the equation by 2
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.
Examples
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Integration
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Limits
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