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x^{2}-10x=13
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^{2}-10x-13=13-13
Subtract 13 from both sides of the equation.
x^{2}-10x-13=0
Subtracting 13 from itself leaves 0.
x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-13\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\left(-13\right)}}{2}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100+52}}{2}
Multiply -4 times -13.
x=\frac{-\left(-10\right)±\sqrt{152}}{2}
Add 100 to 52.
x=\frac{-\left(-10\right)±2\sqrt{38}}{2}
Take the square root of 152.
x=\frac{10±2\sqrt{38}}{2}
The opposite of -10 is 10.
x=\frac{2\sqrt{38}+10}{2}
Now solve the equation x=\frac{10±2\sqrt{38}}{2} when ± is plus. Add 10 to 2\sqrt{38}.
x=\sqrt{38}+5
Divide 10+2\sqrt{38} by 2.
x=\frac{10-2\sqrt{38}}{2}
Now solve the equation x=\frac{10±2\sqrt{38}}{2} when ± is minus. Subtract 2\sqrt{38} from 10.
x=5-\sqrt{38}
Divide 10-2\sqrt{38} by 2.
x=\sqrt{38}+5 x=5-\sqrt{38}
The equation is now solved.
x^{2}-10x=13
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}-10x+\left(-5\right)^{2}=13+\left(-5\right)^{2}
Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-10x+25=13+25
Square -5.
x^{2}-10x+25=38
Add 13 to 25.
\left(x-5\right)^{2}=38
Factor x^{2}-10x+25. 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-5\right)^{2}}=\sqrt{38}
Take the square root of both sides of the equation.
x-5=\sqrt{38} x-5=-\sqrt{38}
Simplify.
x=\sqrt{38}+5 x=5-\sqrt{38}
Add 5 to both sides of the equation.