Problem 5 - College Algebra

Question

What two algebraic methods can be used to fi d the horizontal intercepts of a quadratic function?

Step-by-Step Solution

Answer

Factoring and quadratic formula are the two algebraic methods.

Step 1: Understand the Problem

To find the horizontal intercepts of a quadratic function, we need to solve the equation for when the function's output is zero. These intercepts are also known as the x-intercepts or roots of the quadratic equation.

Step 2: Method 1 - Factoring

If the quadratic equation can be expressed as \( ax^2 + bx + c = 0 \), and if it is factorable, write it in the form of \((px + q)(rx + s) = 0\). Solve for \( x \) by setting each factor equal to zero: \( px + q = 0 \) and \( rx + s = 0 \). Find the values of \( x \) that satisfy these equations.

Step 3: Method 2 - Quadratic Formula

When the quadratic equation \( ax^2 + bx + c = 0 \) cannot be easily factored, use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug in the values of \( a \), \( b \), and \( c \) from the equation, and solve for \( x \) to find the horizontal intercepts.

Key Concepts

Horizontal Intercepts

Horizontal intercepts, also known as x-intercepts or roots, are points where a function crosses the x-axis. For a quadratic function of the form \( ax^2 + bx + c = 0 \), these intercepts are the values of \( x \) which make the function equal to zero. Finding these intercepts is a crucial task because they reveal important features of the function, such as where the curve meets the x-axis.

To find them, you need to solve the equation \( ax^2 + bx + c = 0 \). This involves determining the values of \( x \) that make the output of the function zero. By understanding these intersections, learners can visualize where the graph of the quadratic reaches the x-axis, giving insight into the behavior and shape of the function.

Factoring

Factoring is a popular method for finding the horizontal intercepts of a quadratic function. It involves rewriting the quadratic equation \( ax^2 + bx + c = 0 \) in a product form such as \((px + q)(rx + s) = 0\).

To factor a quadratic, follow these steps:

• Look for two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term \( c \)), and add to \( b \) (the middle coefficient).
• Rewrite the middle term \( bx \) using these two numbers.
• Use grouping to factor the equation by grouping terms and factoring out common factors.
• Once the equation is factored, solve each factor by setting them to zero.

Not all quadratics can be easily factored, so this method works best when the equation has simple coefficients or clear common factors.

Quadratic Formula

The quadratic formula is a universally applicable method for finding the horizontal intercepts of any quadratic equation, particularly when factoring is difficult or impossible.

The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can solve any quadratic equation \( ax^2 + bx + c = 0 \). Here's how to use it:

• Identify the coefficients \( a \), \( b \), and \( c \) from your equation.
• Plug these coefficients into the quadratic formula.
• Solve for \( x \). You might get two solutions, as indicated by the \( \pm \) same term indicating potential for two different horizontal intercepts.
• Don't forget to verify whether \( b^2 - 4ac \) (the discriminant) is positive for real roots. If it's zero, there is one root. If negative, no real solutions exist.

This formula is particularly helpful for complex equations, providing a reliable tool for finding x-intercepts.