Find X-Intercepts Using Zero Product Property - Algebra-1

1. Fundamental Concepts

Quadratic Equations: A quadratic equation is an equation of the form where . Solving a quadratic equation means finding the values of x that satisfy the equation.
x-Intercepts: These are the points where the graph of the quadratic function intersects the x-axis. At these points, the y-value (or function value) is 0. So, finding the x-intercepts of a quadratic function is equivalent to solving the quadratic equation when the function value is 0.
Zero Product Property: If the product of two factors is zero, then at least one of the factors must be zero. In other words, if , then either or (or both). This property is crucial for finding x-intercepts by factoring quadratic equations.

Relationship Between Solutions, Zeros, Roots, and x-Intercepts: For a quadratic function $f(x)=ax^2 + bx + c$ :
- The solutions of the quadratic equation are the same as the roots of the equation.
- The zeros of the function are the values of x for which , which are exactly the solutions/roots of the corresponding quadratic equation.
- The x-intercepts of the graph of the function are the points where x is a zero/root/solution of the quadratic equation. So, the x-coordinates of the x-intercepts are the zeros/roots/solutions.
Using Zero Product Property to Find x-Intercepts: To find the x-intercepts using the Zero Product Property, first, rewrite the quadratic equation in factored form (where ). Then, by the Zero Product Property, set each factor equal to zero and solve for x. The solutions and are the x-coordinates of the x-intercepts, so the x-intercepts are and .

Find the x-intercepts of the quadratic function using the Zero Product Property.

Step 1: Set the function equal to 0 to get the quadratic equation: .
Step 2: Factor the left - hand side. Notice that is a difference of squares, which factors as . So the equation becomes .
Step 3: Apply the Zero Product Property. Set each factor equal to 0: or .
Step 4: Solve for x. For , we get ; for , we get .
Conclusion: The x-intercepts are and .

Find the x-intercepts of the quadratic function using the Zero Product Property.

Step 1: Set , so .
Step 2: Factor the quadratic. We need two numbers that multiply to and add to 5. These numbers are 9 and . So, the equation factors as .
Step 3: Use the Zero Product Property: or .
Step 4: Solve for x: or .
Conclusion: The x-intercepts are and .

Find the x-intercepts of the quadratic function using the Zero Product Property.

Step 1: Set $f(x) = 0$, resulting in $2x^2+8x-24 = 0$.
Step 2: First, factor out the greatest common factor of the terms, which is 2. We get $2(x^2 + 4x-12)=0$. Since $2\neq0$, we can focus on factoring $x^2 + 4x-12$. We need two numbers that multiply to $-12$ and add to 4, which are 6 and $-2$. So, the equation becomes $2(x + 6)(x-2)=0$.
Step 3: Apply the Zero Product Property: $x + 6=0$ or $x-2 = 0$ (since $2\neq0$).
Step 4: Solve for x: $x=-6$ or $x = 2$.
Conclusion: The x-intercepts are $(-6,0)$ and $(2,0)$.

Rewrite the Equation: Start by setting the quadratic function equal to 0, as we are looking for where the function intersects the x-axis (y = 0).
Factor the Quadratic: Factor the quadratic expression into a product of two binomials (if possible). This may involve first factoring out a greatest common factor, recognizing special forms like difference of squares, or finding two numbers that multiply to the constant term and add to the coefficient of the linear term.
Apply the Zero Product Property: Once factored, set each factor equal to 0 and solve each resulting linear equation for x.
Identify x-Intercepts: The solutions for x are the x-coordinates of the x-intercepts, so the x-intercepts are $(x_1,0)$ and $(x_2,0)$ where $x_1$ and $x_2$ are the solutions.
Verification: After finding the potential x-intercepts, substitute them back into the original quadratic function to verify that the function value is 0, ensuring the solutions are correct.