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a+b=-6 ab=8
To solve the equation, factor a^{2}-6a+8 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+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 negative, a and b are both negative. List all such integer pairs that give product 8.
-1-8=-9 -2-4=-6
Calculate the sum for each pair.
a=-4 b=-2
The solution is the pair that gives sum -6.
\left(a-4\right)\left(a-2\right)
Rewrite factored expression \left(a+a\right)\left(a+b\right) using the obtained values.
a=4 a=2
To find equation solutions, solve a-4=0 and a-2=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 a^{2}+aa+ba+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 negative, a and b are both negative. List all such integer pairs that give product 8.
-1-8=-9 -2-4=-6
Calculate the sum for each pair.
a=-4 b=-2
The solution is the pair that gives sum -6.
\left(a^{2}-4a\right)+\left(-2a+8\right)
Rewrite a^{2}-6a+8 as \left(a^{2}-4a\right)+\left(-2a+8\right).
a\left(a-4\right)-2\left(a-4\right)
Factor out a in the first and -2 in the second group.
\left(a-4\right)\left(a-2\right)
Factor out common term a-4 by using distributive property.
a=4 a=2
To find equation solutions, solve a-4=0 and a-2=0.
a^{2}-6a+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.
a=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{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}.
a=\frac{-\left(-6\right)±\sqrt{36-4\times 8}}{2}
Square -6.
a=\frac{-\left(-6\right)±\sqrt{36-32}}{2}
Multiply -4 times 8.
a=\frac{-\left(-6\right)±\sqrt{4}}{2}
Add 36 to -32.
a=\frac{-\left(-6\right)±2}{2}
Take the square root of 4.
a=\frac{6±2}{2}
The opposite of -6 is 6.
a=\frac{8}{2}
Now solve the equation a=\frac{6±2}{2} when ± is plus. Add 6 to 2.
a=4
Divide 8 by 2.
a=\frac{4}{2}
Now solve the equation a=\frac{6±2}{2} when ± is minus. Subtract 2 from 6.
a=2
Divide 4 by 2.
a=4 a=2
The equation is now solved.
a^{2}-6a+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.
a^{2}-6a+8-8=-8
Subtract 8 from both sides of the equation.
a^{2}-6a=-8
Subtracting 8 from itself leaves 0.
a^{2}-6a+\left(-3\right)^{2}=-8+\left(-3\right)^{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.
a^{2}-6a+9=-8+9
Square -3.
a^{2}-6a+9=1
Add -8 to 9.
\left(a-3\right)^{2}=1
Factor a^{2}-6a+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(a-3\right)^{2}}=\sqrt{1}
Take the square root of both sides of the equation.
a-3=1 a-3=-1
Simplify.
a=4 a=2
Add 3 to 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 = 2 s = 3 + 1 = 4
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.