Solve for x
x=1
x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667
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Polynomial
5 problems similar to:
\frac { x ^ { 2 } - 6 x + 9 } { 2 } - x + x ^ { 2 } = x - ( x - 2 )
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x^{2}-6x+9-2x+2x^{2}=2x-2\left(x-2\right)
Multiply both sides of the equation by 2.
x^{2}-8x+9+2x^{2}=2x-2\left(x-2\right)
Combine -6x and -2x to get -8x.
3x^{2}-8x+9=2x-2\left(x-2\right)
Combine x^{2} and 2x^{2} to get 3x^{2}.
3x^{2}-8x+9=2x-2x+4
Use the distributive property to multiply -2 by x-2.
3x^{2}-8x+9=4
Combine 2x and -2x to get 0.
3x^{2}-8x+9-4=0
Subtract 4 from both sides.
3x^{2}-8x+5=0
Subtract 4 from 9 to get 5.
a+b=-8 ab=3\times 5=15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+5. To find a and b, set up a system to be solved.
-1,-15 -3,-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 15.
-1-15=-16 -3-5=-8
Calculate the sum for each pair.
a=-5 b=-3
The solution is the pair that gives sum -8.
\left(3x^{2}-5x\right)+\left(-3x+5\right)
Rewrite 3x^{2}-8x+5 as \left(3x^{2}-5x\right)+\left(-3x+5\right).
x\left(3x-5\right)-\left(3x-5\right)
Factor out x in the first and -1 in the second group.
\left(3x-5\right)\left(x-1\right)
Factor out common term 3x-5 by using distributive property.
x=\frac{5}{3} x=1
To find equation solutions, solve 3x-5=0 and x-1=0.
x^{2}-6x+9-2x+2x^{2}=2x-2\left(x-2\right)
Multiply both sides of the equation by 2.
x^{2}-8x+9+2x^{2}=2x-2\left(x-2\right)
Combine -6x and -2x to get -8x.
3x^{2}-8x+9=2x-2\left(x-2\right)
Combine x^{2} and 2x^{2} to get 3x^{2}.
3x^{2}-8x+9=2x-2x+4
Use the distributive property to multiply -2 by x-2.
3x^{2}-8x+9=4
Combine 2x and -2x to get 0.
3x^{2}-8x+9-4=0
Subtract 4 from both sides.
3x^{2}-8x+5=0
Subtract 4 from 9 to get 5.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\times 5}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -8 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\times 3\times 5}}{2\times 3}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64-12\times 5}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-8\right)±\sqrt{64-60}}{2\times 3}
Multiply -12 times 5.
x=\frac{-\left(-8\right)±\sqrt{4}}{2\times 3}
Add 64 to -60.
x=\frac{-\left(-8\right)±2}{2\times 3}
Take the square root of 4.
x=\frac{8±2}{2\times 3}
The opposite of -8 is 8.
x=\frac{8±2}{6}
Multiply 2 times 3.
x=\frac{10}{6}
Now solve the equation x=\frac{8±2}{6} when ± is plus. Add 8 to 2.
x=\frac{5}{3}
Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.
x=\frac{6}{6}
Now solve the equation x=\frac{8±2}{6} when ± is minus. Subtract 2 from 8.
x=1
Divide 6 by 6.
x=\frac{5}{3} x=1
The equation is now solved.
x^{2}-6x+9-2x+2x^{2}=2x-2\left(x-2\right)
Multiply both sides of the equation by 2.
x^{2}-8x+9+2x^{2}=2x-2\left(x-2\right)
Combine -6x and -2x to get -8x.
3x^{2}-8x+9=2x-2\left(x-2\right)
Combine x^{2} and 2x^{2} to get 3x^{2}.
3x^{2}-8x+9=2x-2x+4
Use the distributive property to multiply -2 by x-2.
3x^{2}-8x+9=4
Combine 2x and -2x to get 0.
3x^{2}-8x=4-9
Subtract 9 from both sides.
3x^{2}-8x=-5
Subtract 9 from 4 to get -5.
\frac{3x^{2}-8x}{3}=-\frac{5}{3}
Divide both sides by 3.
x^{2}-\frac{8}{3}x=-\frac{5}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-\frac{5}{3}+\left(-\frac{4}{3}\right)^{2}
Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{5}{3}+\frac{16}{9}
Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{1}{9}
Add -\frac{5}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{4}{3}\right)^{2}=\frac{1}{9}
Factor x^{2}-\frac{8}{3}x+\frac{16}{9}. 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{4}{3}\right)^{2}}=\sqrt{\frac{1}{9}}
Take the square root of both sides of the equation.
x-\frac{4}{3}=\frac{1}{3} x-\frac{4}{3}=-\frac{1}{3}
Simplify.
x=\frac{5}{3} x=1
Add \frac{4}{3} to both sides of the equation.
Examples
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{ 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}