Skip to main content
Solve for x
Tick mark Image
Graph

Similar Problems from Web Search

Share

x^{2}-13x+27=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(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 27}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and 27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-13\right)±\sqrt{169-4\times 27}}{2}
Square -13.
x=\frac{-\left(-13\right)±\sqrt{169-108}}{2}
Multiply -4 times 27.
x=\frac{-\left(-13\right)±\sqrt{61}}{2}
Add 169 to -108.
x=\frac{13±\sqrt{61}}{2}
The opposite of -13 is 13.
x=\frac{\sqrt{61}+13}{2}
Now solve the equation x=\frac{13±\sqrt{61}}{2} when ± is plus. Add 13 to \sqrt{61}.
x=\frac{13-\sqrt{61}}{2}
Now solve the equation x=\frac{13±\sqrt{61}}{2} when ± is minus. Subtract \sqrt{61} from 13.
x=\frac{\sqrt{61}+13}{2} x=\frac{13-\sqrt{61}}{2}
The equation is now solved.
x^{2}-13x+27=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}-13x+27-27=-27
Subtract 27 from both sides of the equation.
x^{2}-13x=-27
Subtracting 27 from itself leaves 0.
x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-27+\left(-\frac{13}{2}\right)^{2}
Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-13x+\frac{169}{4}=-27+\frac{169}{4}
Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-13x+\frac{169}{4}=\frac{61}{4}
Add -27 to \frac{169}{4}.
\left(x-\frac{13}{2}\right)^{2}=\frac{61}{4}
Factor x^{2}-13x+\frac{169}{4}. 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-\frac{13}{2}\right)^{2}}=\sqrt{\frac{61}{4}}
Take the square root of both sides of the equation.
x-\frac{13}{2}=\frac{\sqrt{61}}{2} x-\frac{13}{2}=-\frac{\sqrt{61}}{2}
Simplify.
x=\frac{\sqrt{61}+13}{2} x=\frac{13-\sqrt{61}}{2}
Add \frac{13}{2} to both sides of the equation.
x ^ 2 -13x +27 = 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 = 13 rs = 27
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 = \frac{13}{2} - u s = \frac{13}{2} + u
Two numbers r and s sum up to 13 exactly when the average of the two numbers is \frac{1}{2}*13 = \frac{13}{2}. 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>
(\frac{13}{2} - u) (\frac{13}{2} + u) = 27
To solve for unknown quantity u, substitute these in the product equation rs = 27
\frac{169}{4} - u^2 = 27
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 27-\frac{169}{4} = -\frac{61}{4}
Simplify the expression by subtracting \frac{169}{4} on both sides
u^2 = \frac{61}{4} u = \pm\sqrt{\frac{61}{4}} = \pm \frac{\sqrt{61}}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{13}{2} - \frac{\sqrt{61}}{2} = 2.595 s = \frac{13}{2} + \frac{\sqrt{61}}{2} = 10.405
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.