Solve (x-5)^2(x+2)/x-5=0

$$\frac { ( x - 5 ) ^ { 2 } ( x + 2 ) } { x - 5 } = 0$$

Solve for x

$$x=-2$$

Solution Steps

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by x-5.

$$\left(x-5\right)^{2}\left(x+2\right)=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$\left(x^{2}-10x+25\right)\left(x+2\right)=0$$

Use the distributive property to multiply x^{2}-10x+25 by x+2 and combine like terms.

$$x^{3}-8x^{2}+5x+50=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 50 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±50,±25,±10,±5,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-8x^{2}+5x+50 by x+2 to get x^{2}-10x+25. Solve the equation where the result equals to 0.

$$x^{2}-10x+25=0$$

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, -10 for b, and 25 for c in the quadratic formula.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\times 25}}{2}$$

Do the calculations.

$$x=\frac{10±0}{2}$$

Solutions are the same.

$$x=5$$

Remove the values that the variable cannot be equal to.

$$x=-2$$

List all found solutions.

$$x=-2 x=5$$

Variable x cannot be equal to 5.

$$x=-2$$