How to use the zero product property?
Understand the Problem
The question is asking for an explanation of how to apply the zero product property in mathematics, which states that if the product of two expressions equals zero, then at least one of the expressions must be zero. This is often used to solve quadratic equations.
Answer
The solutions are $x = 2$ and $x = 3$.
Answer for screen readers
The solutions to the quadratic equation $x^2 - 5x + 6 = 0$ are $x = 2$ and $x = 3$.
Steps to Solve
- Identify the factors of the equation
Start with a quadratic equation in the form $ax^2 + bx + c = 0$. You need to factor the quadratic expression on the left side. For example, if you have $x^2 - 5x + 6 = 0$, you would look for two numbers that multiply to $6$ (the constant term) and add to $-5$ (the coefficient of $x$).
- Apply the zero product property
Once you have factored the equation into two binomials, for example $(x-2)(x-3) = 0$, you can apply the zero product property. This states that if the product of two factors is zero, then at least one of the factors must be zero.
- Set each factor equal to zero
Set each of the factors equal to zero: $$ x - 2 = 0 $$ $$ x - 3 = 0 $$
- Solve for the variable
Solve each equation for $x$: From $x - 2 = 0$, we find $x = 2$.
From $x - 3 = 0$, we find $x = 3$.
- State the solutions
Conclude by stating the solutions to the original quadratic equation. In this case, they are $x = 2$ and $x = 3$.
The solutions to the quadratic equation $x^2 - 5x + 6 = 0$ are $x = 2$ and $x = 3$.
More Information
The zero product property is a fundamental concept in algebra that allows us to solve quadratic equations efficiently. Understanding this property helps in solving more complex equations that can be factored.
Tips
- Forgetting to set each factor equal to zero can lead to missing potential solutions.
- Making errors while factoring can lead to incorrect equations, so it’s important to double-check the factorization process.
