If $(x+ 1) (x + 2) (x + 3) (x + 4) =24$, then which of the following is true?
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Zero is one of the roots of the equation
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The given equation has no roots
- The given equation has exactly $24$ roots
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The given equation is an identity
Rearrange the expression as ((x+1)(x+4)) * ((x+2)(x+3)) = 24, which is (x^2 + 5x + 4)(x^2 + 5x + 6) = 24. Let u = x^2 + 5x, then (u+4)(u+6) = 24, so u^2 + 10u + 24 = 24, leading to u(u+10) = 0. If u = 0, x^2 + 5x = 0, so x(x+5) = 0, meaning 0 is a root.
To solve the equation, we group the first two and last two factors to get (x^2 + 5x + 4)(x + 2)(x + 1) = 24. Multiplying (x + 2) and (x + 1) gives the equation (x^2 + 5x + 4)(x^2 + 5x + 6) = 24. Let y = x^2 + 5x, which yields the quadratic (y + 4)(y + 6) = 24, simplifying to y^2 + 10y = 0. Substituting back gives x(x + 5) = 0, proving zero is a root.