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Solve for x (complex solution)
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x\left(x^{2}+2x+1\right)=4
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
x^{3}+2x^{2}+x=4
Use the distributive property to multiply x by x^{2}+2x+1.
x^{3}+2x^{2}+x-4=0
Subtract 4 from both sides.
±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{2}+3x+4=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+2x^{2}+x-4 by x-1 to get x^{2}+3x+4. Solve the equation where the result equals to 0.
x=\frac{-3±\sqrt{3^{2}-4\times 1\times 4}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 3 for b, and 4 for c in the quadratic formula.
x=\frac{-3±\sqrt{-7}}{2}
Do the calculations.
x=\frac{-\sqrt{7}i-3}{2} x=\frac{-3+\sqrt{7}i}{2}
Solve the equation x^{2}+3x+4=0 when ± is plus and when ± is minus.
x=1 x=\frac{-\sqrt{7}i-3}{2} x=\frac{-3+\sqrt{7}i}{2}
List all found solutions.
x\left(x^{2}+2x+1\right)=4
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
x^{3}+2x^{2}+x=4
Use the distributive property to multiply x by x^{2}+2x+1.
x^{3}+2x^{2}+x-4=0
Subtract 4 from both sides.
±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{2}+3x+4=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+2x^{2}+x-4 by x-1 to get x^{2}+3x+4. Solve the equation where the result equals to 0.
x=\frac{-3±\sqrt{3^{2}-4\times 1\times 4}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 3 for b, and 4 for c in the quadratic formula.
x=\frac{-3±\sqrt{-7}}{2}
Do the calculations.
x\in \emptyset
Since the square root of a negative number is not defined in the real field, there are no solutions.
x=1
List all found solutions.