Problem 69 - Elementary Algebra

Question

Problem: Find the \(x\)-intercepts of the graph of \(y=x^{2}+2 x-24\) Incorrect Answer: \(0=(x-6)(x+4)\) $$ \begin{array}{rrr} x-6=0 & \text { or } & x+4=0 \\ +6+6 & & -4=-4 \\ \hline x+0=6 & x+0=-4 \\ x=6 & x=-4 \end{array} $$ The \(x\)-intercepts are \((6,0)\) and \((-4,0)\).

Step-by-Step Solution

Answer

The x-intercepts are (-6, 0) and (4, 0).

Step 1: Identify the equation

Start with the given quadratic equation: \( y = x^2 + 2x - 24 \)

Step 2: Set y to zero

To find the x-intercepts, set \( y = 0 \) in the equation: \( 0 = x^2 + 2x - 24 \)

Step 3: Factor the quadratic equation

Rewrite the equation by factoring: \( 0 = (x + 6)(x - 4) \)

Step 4: Solve for x

Set each factor equal to zero: \( x + 6 = 0 \) or \( x - 4 = 0 \).Then solve for x:\( x = -6 \) or \( x = 4 \)

Step 5: Identify the x-intercepts

The x-intercepts are the points where the graph crosses the x-axis. They are at: (-6, 0) and (4, 0)

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations form a parabola when graphed on the coordinate plane. The main goal with quadratic equations is to find the value of \(x\) that makes the equation true, which are known as the solutions or roots.

In our example, the equation given is \(y = x^2 + 2x - 24\). To find the \(x\)-intercepts, we need to set \(y = 0\), giving us the equation \(0 = x^2 + 2x - 24\).

Identifying and solving quadratic equations is a crucial skill in algebra. Often, solving them involves factoring the polynomial.

Factoring Polynomials

Factoring is a method used to break down polynomials into simpler factors that, when multiplied together, give the original polynomial equation. In the quadratic equation \(0 = x^2 + 2x - 24\), we need to find two numbers that multiply to \(-24\) (the constant term) and add up to \(2\) (the coefficient of the middle term).

These two numbers are \(6\) and \(-4\). So, we can factor the quadratic as \((x + 6)(x - 4)\). This step transforms the equation into a product of two binomials set to zero: \(0 = (x + 6)(x - 4)\).

Factoring polynomials makes finding solutions much easier, as we can now solve each factor individually.

Solving for x

Once we have factored the polynomial to \(0 = (x + 6)(x - 4)\), the next step is to solve for \(x\).

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This allows us to set each factor equal to zero and solve separately:

• \(x + 6 = 0\)
• \(x - 4 = 0\)

Solving these equations provides the solutions \(x = -6\) and \(x = 4\).

These solutions are the x-intercepts of the graph of the original quadratic equation, meaning the points where the graph crosses the x-axis are \((-6, 0)\) and \((4, 0)\).