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x^{2}-4x+2+3x=4
Add 3x to both sides.
x^{2}-x+2=4
Combine -4x and 3x to get -x.
x^{2}-x+2-4=0
Subtract 4 from both sides.
x^{2}-x-2=0
Subtract 4 from 2 to get -2.
a+b=-1 ab=-2
To solve the equation, factor x^{2}-x-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 negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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}-4x+2+3x=4
Add 3x to both sides.
x^{2}-x+2=4
Combine -4x and 3x to get -x.
x^{2}-x+2-4=0
Subtract 4 from both sides.
x^{2}-x-2=0
Subtract 4 from 2 to get -2.
a+b=-1 ab=1\left(-2\right)=-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 negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.
\left(x^{2}-2x\right)+\left(x-2\right)
Rewrite x^{2}-x-2 as \left(x^{2}-2x\right)+\left(x-2\right).
x\left(x-2\right)+x-2
Factor out x in x^{2}-2x.
\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}-4x+2+3x=4
Add 3x to both sides.
x^{2}-x+2=4
Combine -4x and 3x to get -x.
x^{2}-x+2-4=0
Subtract 4 from both sides.
x^{2}-x-2=0
Subtract 4 from 2 to get -2.
x=\frac{-\left(-1\right)±\sqrt{1-4\left(-2\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1+8}}{2}
Multiply -4 times -2.
x=\frac{-\left(-1\right)±\sqrt{9}}{2}
Add 1 to 8.
x=\frac{-\left(-1\right)±3}{2}
Take the square root of 9.
x=\frac{1±3}{2}
The opposite of -1 is 1.
x=\frac{4}{2}
Now solve the equation x=\frac{1±3}{2} when ± is plus. Add 1 to 3.
x=2
Divide 4 by 2.
x=-\frac{2}{2}
Now solve the equation x=\frac{1±3}{2} when ± is minus. Subtract 3 from 1.
x=-1
Divide -2 by 2.
x=2 x=-1
The equation is now solved.
x^{2}-4x+2+3x=4
Add 3x to both sides.
x^{2}-x+2=4
Combine -4x and 3x to get -x.
x^{2}-x=4-2
Subtract 2 from both sides.
x^{2}-x=2
Subtract 2 from 4 to get 2.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=2+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=2+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{9}{4}
Add 2 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}
Factor x^{2}-x+\frac{1}{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{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{3}{2} x-\frac{1}{2}=-\frac{3}{2}
Simplify.
x=2 x=-1
Add \frac{1}{2} to both sides of the equation.