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x^{2}-6x+45=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(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 45}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-6\right)±\sqrt{36-4\times 45}}{2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36-180}}{2}
Multiply -4 times 45.
x=\frac{-\left(-6\right)±\sqrt{-144}}{2}
Add 36 to -180.
x=\frac{-\left(-6\right)±12i}{2}
Take the square root of -144.
x=\frac{6±12i}{2}
The opposite of -6 is 6.
x=\frac{6+12i}{2}
Now solve the equation x=\frac{6±12i}{2} when ± is plus. Add 6 to 12i.
x=3+6i
Divide 6+12i by 2.
x=\frac{6-12i}{2}
Now solve the equation x=\frac{6±12i}{2} when ± is minus. Subtract 12i from 6.
x=3-6i
Divide 6-12i by 2.
x=3+6i x=3-6i
The equation is now solved.
x^{2}-6x+45=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.
x^{2}-6x+45-45=-45
Subtract 45 from both sides of the equation.
x^{2}-6x=-45
Subtracting 45 from itself leaves 0.
x^{2}-6x+\left(-3\right)^{2}=-45+\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.
x^{2}-6x+9=-45+9
Square -3.
x^{2}-6x+9=-36
Add -45 to 9.
\left(x-3\right)^{2}=-36
Factor x^{2}-6x+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(x-3\right)^{2}}=\sqrt{-36}
Take the square root of both sides of the equation.
x-3=6i x-3=-6i
Simplify.
x=3+6i x=3-6i
Add 3 to both sides of the equation.
x ^ 2 -6x +45 = 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 = 45
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) = 45
To solve for unknown quantity u, substitute these in the product equation rs = 45
9 - u^2 = 45
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 45-9 = 36
Simplify the expression by subtracting 9 on both sides
u^2 = -36 u = \pm\sqrt{-36} = \pm 6i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =3 - 6i s = 3 + 6i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.