Problem 98 - Algebra and Trigonometry

Question

Solve each equation. $$x(x-2)^{3}-35(x-2)^{2}=0$$

Step-by-Step Solution

Answer

The roots of the equation are \(x=-5\), \(x=2\) and \(x=7\).

Step 1: Simplifying the Equation

The equation \(x(x-2)^{3}-35(x-2)^{2}=0\) can be simplified by factoring out \((x-2)^{2}\) from each term which will result in \((x-2)^{2}(x(x-2) - 35)=0\). Thus, rewrite the equation as \((x-2)^{2}(x^2-2x-35)=0\).

Step 2: Further Factorizing the Equation

The quadratic equation inside the parenthesis can be further factored. So, factorize \(x^2-2x-35\) which will result in \((x-7)(x+5)\). Now, rewrite the entire equation as \((x-2)^2(x-7)(x+5)=0\).

Step 3: Find the Roots of the Equation

Set each factor equal to zero and solve for x. This gives the roots of the equation as \(x-2 = 0\), \(x-7=0\) and \(x+5=0\). Solving each yields \(x=2\), \(x=7\) and \(x=-5\).