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x^{2}-10x+25=0
Divide both sides by 2.
a+b=-10 ab=1\times 25=25
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+25. To find a and b, set up a system to be solved.
-1,-25 -5,-5
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 25.
-1-25=-26 -5-5=-10
Calculate the sum for each pair.
a=-5 b=-5
The solution is the pair that gives sum -10.
\left(x^{2}-5x\right)+\left(-5x+25\right)
Rewrite x^{2}-10x+25 as \left(x^{2}-5x\right)+\left(-5x+25\right).
x\left(x-5\right)-5\left(x-5\right)
Factor out x in the first and -5 in the second group.
\left(x-5\right)\left(x-5\right)
Factor out common term x-5 by using distributive property.
\left(x-5\right)^{2}
Rewrite as a binomial square.
x=5
To find equation solution, solve x-5=0.
2x^{2}-20x+50=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(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 2\times 50}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -20 for b, and 50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-20\right)±\sqrt{400-4\times 2\times 50}}{2\times 2}
Square -20.
x=\frac{-\left(-20\right)±\sqrt{400-8\times 50}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-20\right)±\sqrt{400-400}}{2\times 2}
Multiply -8 times 50.
x=\frac{-\left(-20\right)±\sqrt{0}}{2\times 2}
Add 400 to -400.
x=-\frac{-20}{2\times 2}
Take the square root of 0.
x=\frac{20}{2\times 2}
The opposite of -20 is 20.
x=\frac{20}{4}
Multiply 2 times 2.
x=5
Divide 20 by 4.
2x^{2}-20x+50=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.
2x^{2}-20x+50-50=-50
Subtract 50 from both sides of the equation.
2x^{2}-20x=-50
Subtracting 50 from itself leaves 0.
\frac{2x^{2}-20x}{2}=-\frac{50}{2}
Divide both sides by 2.
x^{2}+\left(-\frac{20}{2}\right)x=-\frac{50}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-10x=-\frac{50}{2}
Divide -20 by 2.
x^{2}-10x=-25
Divide -50 by 2.
x^{2}-10x+\left(-5\right)^{2}=-25+\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=-25+25
Square -5.
x^{2}-10x+25=0
Add -25 to 25.
\left(x-5\right)^{2}=0
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{0}
Take the square root of both sides of the equation.
x-5=0 x-5=0
Simplify.
x=5 x=5
Add 5 to both sides of the equation.
x=5
The equation is now solved. Solutions are the same.