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(2x-1)(x+5)=0

2 min read Jun 16, 2024
(2x-1)(x+5)=0

Solving the Equation (2x-1)(x+5) = 0

This equation involves a product of two factors that equals zero. This is a fundamental concept in algebra, known as the Zero Product Property.

The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Applying the Zero Product Property

To solve the equation (2x-1)(x+5) = 0, we use the Zero Product Property:

  1. Set each factor equal to zero:

    • 2x - 1 = 0
    • x + 5 = 0

  2. Solve each equation:

    • 2x = 1 => x = 1/2
    • x = -5

Therefore, the solutions to the equation (2x-1)(x+5) = 0 are x = 1/2 and x = -5.

Verifying the Solutions

We can verify these solutions by substituting them back into the original equation:

  • For x = 1/2:

    • (2(1/2) - 1)(1/2 + 5) = (1 - 1)(1/2 + 5) = 0 * (11/2) = 0

  • For x = -5:

    • (2(-5) - 1)(-5 + 5) = (-10 - 1)(0) = -11 * 0 = 0

Since both substitutions result in the equation being true, we have confirmed that x = 1/2 and x = -5 are indeed the solutions.

Conclusion

By applying the Zero Product Property, we were able to efficiently solve the equation (2x-1)(x+5) = 0 and find its two solutions: x = 1/2 and x = -5. This demonstrates the power of factoring and the Zero Product Property in simplifying and solving algebraic equations.

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