Solve for x
x=-1
x = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333
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x^{2}+\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-2x\right)^{2}.
x^{2}+x^{4}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
x^{2}+x^{4}-4x^{3}+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
5x^{2}+x^{4}-4x^{3}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
Combine x^{2} and 4x^{2} to get 5x^{2}.
5x^{2}+x^{4}-4x^{3}=10+x^{2}+2x+1+\left(x^{2}-2x-3\right)^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+\left(x^{2}-2x-3\right)^{2}
Add 10 and 1 to get 11.
5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+x^{4}-4x^{3}-2x^{2}+12x+9
Square x^{2}-2x-3.
5x^{2}+x^{4}-4x^{3}=11-x^{2}+2x+x^{4}-4x^{3}+12x+9
Combine x^{2} and -2x^{2} to get -x^{2}.
5x^{2}+x^{4}-4x^{3}=11-x^{2}+14x+x^{4}-4x^{3}+9
Combine 2x and 12x to get 14x.
5x^{2}+x^{4}-4x^{3}=20-x^{2}+14x+x^{4}-4x^{3}
Add 11 and 9 to get 20.
5x^{2}+x^{4}-4x^{3}-20=-x^{2}+14x+x^{4}-4x^{3}
Subtract 20 from both sides.
5x^{2}+x^{4}-4x^{3}-20+x^{2}=14x+x^{4}-4x^{3}
Add x^{2} to both sides.
6x^{2}+x^{4}-4x^{3}-20=14x+x^{4}-4x^{3}
Combine 5x^{2} and x^{2} to get 6x^{2}.
6x^{2}+x^{4}-4x^{3}-20-14x=x^{4}-4x^{3}
Subtract 14x from both sides.
6x^{2}+x^{4}-4x^{3}-20-14x-x^{4}=-4x^{3}
Subtract x^{4} from both sides.
6x^{2}-4x^{3}-20-14x=-4x^{3}
Combine x^{4} and -x^{4} to get 0.
6x^{2}-4x^{3}-20-14x+4x^{3}=0
Add 4x^{3} to both sides.
6x^{2}-20-14x=0
Combine -4x^{3} and 4x^{3} to get 0.
3x^{2}-10-7x=0
Divide both sides by 2.
3x^{2}-7x-10=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-7 ab=3\left(-10\right)=-30
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-10. To find a and b, set up a system to be solved.
1,-30 2,-15 3,-10 5,-6
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. List all such integer pairs that give product -30.
1-30=-29 2-15=-13 3-10=-7 5-6=-1
Calculate the sum for each pair.
a=-10 b=3
The solution is the pair that gives sum -7.
\left(3x^{2}-10x\right)+\left(3x-10\right)
Rewrite 3x^{2}-7x-10 as \left(3x^{2}-10x\right)+\left(3x-10\right).
x\left(3x-10\right)+3x-10
Factor out x in 3x^{2}-10x.
\left(3x-10\right)\left(x+1\right)
Factor out common term 3x-10 by using distributive property.
x=\frac{10}{3} x=-1
To find equation solutions, solve 3x-10=0 and x+1=0.
x^{2}+\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-2x\right)^{2}.
x^{2}+x^{4}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
x^{2}+x^{4}-4x^{3}+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
5x^{2}+x^{4}-4x^{3}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
Combine x^{2} and 4x^{2} to get 5x^{2}.
5x^{2}+x^{4}-4x^{3}=10+x^{2}+2x+1+\left(x^{2}-2x-3\right)^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+\left(x^{2}-2x-3\right)^{2}
Add 10 and 1 to get 11.
5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+x^{4}-4x^{3}-2x^{2}+12x+9
Square x^{2}-2x-3.
5x^{2}+x^{4}-4x^{3}=11-x^{2}+2x+x^{4}-4x^{3}+12x+9
Combine x^{2} and -2x^{2} to get -x^{2}.
5x^{2}+x^{4}-4x^{3}=11-x^{2}+14x+x^{4}-4x^{3}+9
Combine 2x and 12x to get 14x.
5x^{2}+x^{4}-4x^{3}=20-x^{2}+14x+x^{4}-4x^{3}
Add 11 and 9 to get 20.
5x^{2}+x^{4}-4x^{3}-20=-x^{2}+14x+x^{4}-4x^{3}
Subtract 20 from both sides.
5x^{2}+x^{4}-4x^{3}-20+x^{2}=14x+x^{4}-4x^{3}
Add x^{2} to both sides.
6x^{2}+x^{4}-4x^{3}-20=14x+x^{4}-4x^{3}
Combine 5x^{2} and x^{2} to get 6x^{2}.
6x^{2}+x^{4}-4x^{3}-20-14x=x^{4}-4x^{3}
Subtract 14x from both sides.
6x^{2}+x^{4}-4x^{3}-20-14x-x^{4}=-4x^{3}
Subtract x^{4} from both sides.
6x^{2}-4x^{3}-20-14x=-4x^{3}
Combine x^{4} and -x^{4} to get 0.
6x^{2}-4x^{3}-20-14x+4x^{3}=0
Add 4x^{3} to both sides.
6x^{2}-20-14x=0
Combine -4x^{3} and 4x^{3} to get 0.
6x^{2}-14x-20=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(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 6\left(-20\right)}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -14 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-14\right)±\sqrt{196-4\times 6\left(-20\right)}}{2\times 6}
Square -14.
x=\frac{-\left(-14\right)±\sqrt{196-24\left(-20\right)}}{2\times 6}
Multiply -4 times 6.
x=\frac{-\left(-14\right)±\sqrt{196+480}}{2\times 6}
Multiply -24 times -20.
x=\frac{-\left(-14\right)±\sqrt{676}}{2\times 6}
Add 196 to 480.
x=\frac{-\left(-14\right)±26}{2\times 6}
Take the square root of 676.
x=\frac{14±26}{2\times 6}
The opposite of -14 is 14.
x=\frac{14±26}{12}
Multiply 2 times 6.
x=\frac{40}{12}
Now solve the equation x=\frac{14±26}{12} when ± is plus. Add 14 to 26.
x=\frac{10}{3}
Reduce the fraction \frac{40}{12} to lowest terms by extracting and canceling out 4.
x=-\frac{12}{12}
Now solve the equation x=\frac{14±26}{12} when ± is minus. Subtract 26 from 14.
x=-1
Divide -12 by 12.
x=\frac{10}{3} x=-1
The equation is now solved.
x^{2}+\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-2x\right)^{2}.
x^{2}+x^{4}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
x^{2}+x^{4}-4x^{3}+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
5x^{2}+x^{4}-4x^{3}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}
Combine x^{2} and 4x^{2} to get 5x^{2}.
5x^{2}+x^{4}-4x^{3}=10+x^{2}+2x+1+\left(x^{2}-2x-3\right)^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+\left(x^{2}-2x-3\right)^{2}
Add 10 and 1 to get 11.
5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+x^{4}-4x^{3}-2x^{2}+12x+9
Square x^{2}-2x-3.
5x^{2}+x^{4}-4x^{3}=11-x^{2}+2x+x^{4}-4x^{3}+12x+9
Combine x^{2} and -2x^{2} to get -x^{2}.
5x^{2}+x^{4}-4x^{3}=11-x^{2}+14x+x^{4}-4x^{3}+9
Combine 2x and 12x to get 14x.
5x^{2}+x^{4}-4x^{3}=20-x^{2}+14x+x^{4}-4x^{3}
Add 11 and 9 to get 20.
5x^{2}+x^{4}-4x^{3}+x^{2}=20+14x+x^{4}-4x^{3}
Add x^{2} to both sides.
6x^{2}+x^{4}-4x^{3}=20+14x+x^{4}-4x^{3}
Combine 5x^{2} and x^{2} to get 6x^{2}.
6x^{2}+x^{4}-4x^{3}-14x=20+x^{4}-4x^{3}
Subtract 14x from both sides.
6x^{2}+x^{4}-4x^{3}-14x-x^{4}=20-4x^{3}
Subtract x^{4} from both sides.
6x^{2}-4x^{3}-14x=20-4x^{3}
Combine x^{4} and -x^{4} to get 0.
6x^{2}-4x^{3}-14x+4x^{3}=20
Add 4x^{3} to both sides.
6x^{2}-14x=20
Combine -4x^{3} and 4x^{3} to get 0.
\frac{6x^{2}-14x}{6}=\frac{20}{6}
Divide both sides by 6.
x^{2}+\left(-\frac{14}{6}\right)x=\frac{20}{6}
Dividing by 6 undoes the multiplication by 6.
x^{2}-\frac{7}{3}x=\frac{20}{6}
Reduce the fraction \frac{-14}{6} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{7}{3}x=\frac{10}{3}
Reduce the fraction \frac{20}{6} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=\frac{10}{3}+\left(-\frac{7}{6}\right)^{2}
Divide -\frac{7}{3}, the coefficient of the x term, by 2 to get -\frac{7}{6}. Then add the square of -\frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{10}{3}+\frac{49}{36}
Square -\frac{7}{6} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{169}{36}
Add \frac{10}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{7}{6}\right)^{2}=\frac{169}{36}
Factor x^{2}-\frac{7}{3}x+\frac{49}{36}. 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{7}{6}\right)^{2}}=\sqrt{\frac{169}{36}}
Take the square root of both sides of the equation.
x-\frac{7}{6}=\frac{13}{6} x-\frac{7}{6}=-\frac{13}{6}
Simplify.
x=\frac{10}{3} x=-1
Add \frac{7}{6} 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}