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a+b=8 ab=16
To solve the equation, factor x^{2}+8x+16 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). 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+4\right)\left(x+4\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
\left(x+4\right)^{2}
Rewrite as a binomial square.
x=-4
To find equation solution, solve x+4=0.
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 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.
\left(x+4\right)^{2}
Rewrite as a binomial square.
x=-4
To find equation solution, solve x+4=0.
x^{2}+8x+16=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{-8±\sqrt{8^{2}-4\times 16}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\times 16}}{2}
Square 8.
x=\frac{-8±\sqrt{64-64}}{2}
Multiply -4 times 16.
x=\frac{-8±\sqrt{0}}{2}
x=-\frac{8}{2}
Take the square root of 0.
x=-4
Divide -8 by 2.
\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
Subtract 4 from 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.
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.