In Exercises 101–106, solve each equation.x(x+1)3−42(x+1)2=0 x(x + $$1)^3 - 42(x$$ + $$1)^2 = 0$$

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 105

Capitolo 2, Problema 105

In Exercises 101–106, solve each equation. $x (x + 1)^{3} - 42 (x + 1)^{2} = 0$

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Start by observing the given equation: $x(x + 1)^3 - 42(x + 1)^2 = 0$. Notice that both terms contain a common factor involving $(x + 1)^2$.

Factor out the greatest common factor (GCF), which is $(x + 1)^2$, from the entire equation: $ (x + 1)^2 \left[ x(x + 1) - 42 \right] = 0 $.

Simplify the expression inside the brackets: $x(x + 1) - 42 = x^2 + x - 42$.

Set each factor equal to zero to find the solutions: 1) $(x + 1)^2 = 0$ 2) $x^2 + x - 42 = 0$.

Solve each equation separately: - For $(x + 1)^2 = 0$, solve for $x$. - For $x^2 + x - 42 = 0$, use the quadratic formula or factoring to find the values of $x$.

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