Solve for x
x=1
x=2
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x^{2}+2x+1-5x+1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
x^{2}-3x+1+1=0
Combine 2x and -5x to get -3x.
x^{2}-3x+2=0
Add 1 and 1 to get 2.
a+b=-3 ab=2
To solve the equation, factor x^{2}-3x+2 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.
a=-2 b=-1
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.
\left(x-2\right)\left(x-1\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=2 x=1
To find equation solutions, solve x-2=0 and x-1=0.
x^{2}+2x+1-5x+1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
x^{2}-3x+1+1=0
Combine 2x and -5x to get -3x.
x^{2}-3x+2=0
Add 1 and 1 to get 2.
a+b=-3 ab=1\times 2=2
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+2. To find a and b, set up a system to be solved.
a=-2 b=-1
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.
\left(x^{2}-2x\right)+\left(-x+2\right)
Rewrite x^{2}-3x+2 as \left(x^{2}-2x\right)+\left(-x+2\right).
x\left(x-2\right)-\left(x-2\right)
Factor out x in the first and -1 in the second group.
\left(x-2\right)\left(x-1\right)
Factor out common term x-2 by using distributive property.
x=2 x=1
To find equation solutions, solve x-2=0 and x-1=0.
x^{2}+2x+1-5x+1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
x^{2}-3x+1+1=0
Combine 2x and -5x to get -3x.
x^{2}-3x+2=0
Add 1 and 1 to get 2.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 2}}{2}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-8}}{2}
Multiply -4 times 2.
x=\frac{-\left(-3\right)±\sqrt{1}}{2}
Add 9 to -8.
x=\frac{-\left(-3\right)±1}{2}
Take the square root of 1.
x=\frac{3±1}{2}
The opposite of -3 is 3.
x=\frac{4}{2}
Now solve the equation x=\frac{3±1}{2} when ± is plus. Add 3 to 1.
x=2
Divide 4 by 2.
x=\frac{2}{2}
Now solve the equation x=\frac{3±1}{2} when ± is minus. Subtract 1 from 3.
x=1
Divide 2 by 2.
x=2 x=1
The equation is now solved.
x^{2}+2x+1-5x+1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
x^{2}-3x+1+1=0
Combine 2x and -5x to get -3x.
x^{2}-3x+2=0
Add 1 and 1 to get 2.
x^{2}-3x=-2
Subtract 2 from both sides. Anything subtracted from zero gives its negation.
x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-2+\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-3x+\frac{9}{4}=-2+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-3x+\frac{9}{4}=\frac{1}{4}
Add -2 to \frac{9}{4}.
\left(x-\frac{3}{2}\right)^{2}=\frac{1}{4}
Factor x^{2}-3x+\frac{9}{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{3}{2}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{1}{2} x-\frac{3}{2}=-\frac{1}{2}
Simplify.
x=2 x=1
Add \frac{3}{2} to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}