Solve for x
x=3
x=15
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a+b=-18 ab=45
To solve the equation, factor x^{2}-18x+45 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,-45 -3,-15 -5,-9
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 45.
-1-45=-46 -3-15=-18 -5-9=-14
Calculate the sum for each pair.
a=-15 b=-3
The solution is the pair that gives sum -18.
\left(x-15\right)\left(x-3\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=15 x=3
To find equation solutions, solve x-15=0 and x-3=0.
a+b=-18 ab=1\times 45=45
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+45. To find a and b, set up a system to be solved.
-1,-45 -3,-15 -5,-9
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 45.
-1-45=-46 -3-15=-18 -5-9=-14
Calculate the sum for each pair.
a=-15 b=-3
The solution is the pair that gives sum -18.
\left(x^{2}-15x\right)+\left(-3x+45\right)
Rewrite x^{2}-18x+45 as \left(x^{2}-15x\right)+\left(-3x+45\right).
x\left(x-15\right)-3\left(x-15\right)
Factor out x in the first and -3 in the second group.
\left(x-15\right)\left(x-3\right)
Factor out common term x-15 by using distributive property.
x=15 x=3
To find equation solutions, solve x-15=0 and x-3=0.
x^{2}-18x+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(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 45}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-18\right)±\sqrt{324-4\times 45}}{2}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324-180}}{2}
Multiply -4 times 45.
x=\frac{-\left(-18\right)±\sqrt{144}}{2}
Add 324 to -180.
x=\frac{-\left(-18\right)±12}{2}
Take the square root of 144.
x=\frac{18±12}{2}
The opposite of -18 is 18.
x=\frac{30}{2}
Now solve the equation x=\frac{18±12}{2} when ± is plus. Add 18 to 12.
x=15
Divide 30 by 2.
x=\frac{6}{2}
Now solve the equation x=\frac{18±12}{2} when ± is minus. Subtract 12 from 18.
x=3
Divide 6 by 2.
x=15 x=3
The equation is now solved.
x^{2}-18x+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}-18x+45-45=-45
Subtract 45 from both sides of the equation.
x^{2}-18x=-45
Subtracting 45 from itself leaves 0.
x^{2}-18x+\left(-9\right)^{2}=-45+\left(-9\right)^{2}
Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-18x+81=-45+81
Square -9.
x^{2}-18x+81=36
Add -45 to 81.
\left(x-9\right)^{2}=36
Factor x^{2}-18x+81. 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-9\right)^{2}}=\sqrt{36}
Take the square root of both sides of the equation.
x-9=6 x-9=-6
Simplify.
x=15 x=3
Add 9 to both sides of the equation.
x ^ 2 -18x +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 = 18 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 = 9 - u s = 9 + u
Two numbers r and s sum up to 18 exactly when the average of the two numbers is \frac{1}{2}*18 = 9. 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>
(9 - u) (9 + u) = 45
To solve for unknown quantity u, substitute these in the product equation rs = 45
81 - u^2 = 45
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 45-81 = -36
Simplify the expression by subtracting 81 on both sides
u^2 = 36 u = \pm\sqrt{36} = \pm 6
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =9 - 6 = 3 s = 9 + 6 = 15
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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