Problem 16 - Algebra 1

Question

Find the \(x\) -intercepts of the graph of the equation. $$y=x^{2}-11 x+24$$

Step-by-Step Solution

Answer

The \(x\)-intercepts of the given equation are \(x = 3\) and \(x = 8\).

Step 1: Set up the Equation

Set the equation \(y = x^{2} - 11x + 24\) equal to zero because \(y = 0\) at the \(x\)-intercepts. Hence, the equation becomes: \(x^{2} - 11x + 24 = 0\).

Step 2: Factor the Quadratic Equation

The quadratic equation can be factored as following: \(x^{2} -11x + 24\) = \((x-3)(x-8)\). Here, it's assumed that familiarity with factoring quadratic equations - in this case, finding two numbers that multiply to +24 and add to -11.

Step 3: Use the Zero-Product Property to Solve for \(x\)

From step 2, we have \((x-3)(x-8) = 0\). The zero-product property states that if the product of multiple factors equals zero, then at least one of the factors must be zero. Thus, we set each factor equal to zero and solve for \(x\): \(x - 3 = 0\) which leads to \(x=3\) and \(x - 8 = 0\) which leads to \(x = 8\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill when solving quadratic equations. A quadratic equation typically takes the form \[ ax^2 + bx + c = 0 \] To factor such an equation, you need to break it down into two binomial expressions whose product gives the original quadratic.

The main goal is to find two numbers that:

• Multiply to give the constant term, in this case, 24.
• Add up to give the coefficient of the linear term, in this case, -11.

In our example, we factor as \[(x-3)(x-8)\] .

This means that the numbers 3 and 8 multiply to 24, and -3 and -8 add up to -11. Remember, factoring is like un-multiplying that quadratic expression back into its factors. It’s often a trial-and-error process unless it’s a perfect square or fits a special factoring pattern.

Zero-Product Property

The zero-product property is a fundamental rule in algebra that you can use once you've factored your quadratic equation.
According to this property, if the product of two or more numbers is zero, then at least one of the numbers must be zero.

In practical terms, if \[(a)(b) = 0\] then either \[a = 0\] or \[b = 0\] .

For our equation, \[(x-3)(x-8) = 0\] , it implies

• Either \[(x-3) = 0\] leading to \[x=3\] , or
• \[(x-8) = 0\] leading to \[x=8\] .

This property is essential because it turns a complex quadratic into simple linear equations that are easy to solve.

x-intercepts

Finding the x-intercepts of a function like \[y = x^2 - 11x + 24\] involves determining the points where the graph of the equation crosses the x-axis.

At the x-intercepts, the value of \[y\] is 0. That's why we set the original equation equal to zero before solving for \[x\] .

After factoring and applying the zero-product property, we found the solutions \[x=3 \] and \[x=8\] , which are the x-intercepts of the graph.

These x-values are where the graph touches or crosses the x-axis, showing that the corresponding \[y\] values are zero at these points. This concept not only helps in graphing a quadratic equation but also provides insights into the number and location of roots the equation has.