a+b=6 ab=8
To solve the equation, factor b^{2}+6b+8 using formula b^{2}+\left(a+b\right)b+ab=\left(b+a\right)\left(b+b\right). To find a and b, set up a system to be solved.
1,8 2,4
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 8.
1+8=9 2+4=6
Calculate the sum for each pair.
a=2 b=4
The solution is the pair that gives sum 6.
\left(b+2\right)\left(b+4\right)
Rewrite factored expression \left(b+a\right)\left(b+b\right) using the obtained values.
b=-2 b=-4
To find equation solutions, solve b+2=0 and b+4=0.
a+b=6 ab=1\times 8=8
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as b^{2}+ab+bb+8. To find a and b, set up a system to be solved.
1,8 2,4
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 8.
1+8=9 2+4=6
Calculate the sum for each pair.
a=2 b=4
The solution is the pair that gives sum 6.
\left(b^{2}+2b\right)+\left(4b+8\right)
Rewrite b^{2}+6b+8 as \left(b^{2}+2b\right)+\left(4b+8\right).
b\left(b+2\right)+4\left(b+2\right)
Factor out b in the first and 4 in the second group.
\left(b+2\right)\left(b+4\right)
Factor out common term b+2 by using distributive property.
b=-2 b=-4
To find equation solutions, solve b+2=0 and b+4=0.
b^{2}+6b+8=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.
b=\frac{-6±\sqrt{6^{2}-4\times 8}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
b=\frac{-6±\sqrt{36-4\times 8}}{2}
Square 6.
b=\frac{-6±\sqrt{36-32}}{2}
Multiply -4 times 8.
b=\frac{-6±\sqrt{4}}{2}
Add 36 to -32.
b=\frac{-6±2}{2}
Take the square root of 4.
b=-\frac{4}{2}
Now solve the equation b=\frac{-6±2}{2} when ± is plus. Add -6 to 2.
b=-2
Divide -4 by 2.
b=-\frac{8}{2}
Now solve the equation b=\frac{-6±2}{2} when ± is minus. Subtract 2 from -6.
b=-4
Divide -8 by 2.
b=-2 b=-4
The equation is now solved.
b^{2}+6b+8=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.
b^{2}+6b+8-8=-8
Subtract 8 from both sides of the equation.
b^{2}+6b=-8
Subtracting 8 from itself leaves 0.
b^{2}+6b+3^{2}=-8+3^{2}
Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
b^{2}+6b+9=-8+9
Square 3.
b^{2}+6b+9=1
Add -8 to 9.
\left(b+3\right)^{2}=1
Factor b^{2}+6b+9. 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(b+3\right)^{2}}=\sqrt{1}
Take the square root of both sides of the equation.
b+3=1 b+3=-1
Simplify.
b=-2 b=-4
Subtract 3 from both sides of the equation.
x ^ 2 +6x +8 = 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 = -6 rs = 8
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 = -3 - u s = -3 + u
Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. 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>
(-3 - u) (-3 + u) = 8
To solve for unknown quantity u, substitute these in the product equation rs = 8
9 - u^2 = 8
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 8-9 = -1
Simplify the expression by subtracting 9 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 =-3 - 1 = -4 s = -3 + 1 = -2
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
