Solve x(x-8)(x-4)^2=225

$$x ( x - 8 ) ( x - 4 ) ^ { 2 } = 225$$

Solve for x (complex solution)

$$x=-1$$

$$x=9$$

$$x=4-3i$$

$$x=4+3i$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x\left(x-8\right)\left(x^{2}-8x+16\right)=225$$

Use the distributive property to multiply x by x-8.

$$\left(x^{2}-8x\right)\left(x^{2}-8x+16\right)=225$$

Use the distributive property to multiply x^{2}-8x by x^{2}-8x+16 and combine like terms.

$$x^{4}-16x^{3}+80x^{2}-128x=225$$

Subtract 225 from both sides.

$$x^{4}-16x^{3}+80x^{2}-128x-225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-16x^{3}+80x^{2}-128x-225 by x+1 to get x^{3}-17x^{2}+97x-225. Solve the equation where the result equals to 0.

$$x^{3}-17x^{2}+97x-225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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.

$$x=9$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-17x^{2}+97x-225 by x-9 to get x^{2}-8x+25. Solve the equation where the result equals to 0.

$$x^{2}-8x+25=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -8 for b, and 25 for c in the quadratic formula.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\times 25}}{2}$$

Do the calculations.

$$x=\frac{8±\sqrt{-36}}{2}$$

Solve the equation x^{2}-8x+25=0 when ± is plus and when ± is minus.

$$x=4-3i x=4+3i$$

List all found solutions.

$$x=-1 x=9 x=4-3i x=4+3i$$

Solve for x

$$x=9$$

$$x=-1$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x\left(x-8\right)\left(x^{2}-8x+16\right)=225$$

Use the distributive property to multiply x by x-8.

$$\left(x^{2}-8x\right)\left(x^{2}-8x+16\right)=225$$

Use the distributive property to multiply x^{2}-8x by x^{2}-8x+16 and combine like terms.

$$x^{4}-16x^{3}+80x^{2}-128x=225$$

Subtract 225 from both sides.

$$x^{4}-16x^{3}+80x^{2}-128x-225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-16x^{3}+80x^{2}-128x-225 by x+1 to get x^{3}-17x^{2}+97x-225. Solve the equation where the result equals to 0.

$$x^{3}-17x^{2}+97x-225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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.

$$x=9$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-17x^{2}+97x-225 by x-9 to get x^{2}-8x+25. Solve the equation where the result equals to 0.

$$x^{2}-8x+25=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -8 for b, and 25 for c in the quadratic formula.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\times 25}}{2}$$

Do the calculations.

$$x=\frac{8±\sqrt{-36}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=-1 x=9$$