Problem

Solve \(2x^{2}-10x=0\) by first factoring out the greatest common monomial. Write both zero-product equations and solve each one so no solution is omitted.

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What this problem is really about

A variable greatest common factor is itself a zero-producing factor, so it must not be divided away. We’ll extract the greatest common monomial, apply the zero-product property to every remaining factor, solve both branches, and verify the complete set in the original equation.

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2x(x − 5) = 0 gives x = 2 or x = 5. The factor 2x equals zero when x = 0, not 2.

Factor out 2x: 2x(x − 5) = 0. Thus x = 0 or x = 5, and both values satisfy the original equation.

Dividing by x gives 2x − 10 = 0, so x = 5. This loses the solution x = 0.

Factoring out 2 gives 2(x² − 5x) = 0. The remaining shared factor x has not been factored, so the solutions are incomplete.

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Solve \(2x^{2}-10x=0\) by first factoring out the greatest common monomial. Write both zero-product equations and solve each one so no solution is omitted.

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