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m^{2}-6m+18=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.
m=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 18}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-6\right)±\sqrt{36-4\times 18}}{2}
Square -6.
m=\frac{-\left(-6\right)±\sqrt{36-72}}{2}
Multiply -4 times 18.
m=\frac{-\left(-6\right)±\sqrt{-36}}{2}
Add 36 to -72.
m=\frac{-\left(-6\right)±6i}{2}
Take the square root of -36.
m=\frac{6±6i}{2}
The opposite of -6 is 6.
m=\frac{6+6i}{2}
Now solve the equation m=\frac{6±6i}{2} when ± is plus. Add 6 to 6i.
m=3+3i
Divide 6+6i by 2.
m=\frac{6-6i}{2}
Now solve the equation m=\frac{6±6i}{2} when ± is minus. Subtract 6i from 6.
m=3-3i
Divide 6-6i by 2.
m=3+3i m=3-3i
The equation is now solved.
m^{2}-6m+18=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.
m^{2}-6m+18-18=-18
Subtract 18 from both sides of the equation.
m^{2}-6m=-18
Subtracting 18 from itself leaves 0.
m^{2}-6m+\left(-3\right)^{2}=-18+\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.
m^{2}-6m+9=-18+9
Square -3.
m^{2}-6m+9=-9
Add -18 to 9.
\left(m-3\right)^{2}=-9
Factor m^{2}-6m+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(m-3\right)^{2}}=\sqrt{-9}
Take the square root of both sides of the equation.
m-3=3i m-3=-3i
Simplify.
m=3+3i m=3-3i
Add 3 to both sides of the equation.
x ^ 2 -6x +18 = 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 = 18
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) = 18
To solve for unknown quantity u, substitute these in the product equation rs = 18
9 - u^2 = 18
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 18-9 = 9
Simplify the expression by subtracting 9 on both sides
u^2 = -9 u = \pm\sqrt{-9} = \pm 3i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =3 - 3i s = 3 + 3i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.