Problem 5 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Factoring the quadratic or using the quadratic formula.

Step 1: Understanding the Quadratic Function

A quadratic function is typically expressed in the standard form: \[ f(x) = ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Finding the horizontal intercepts involves determining the values of \( x \) where \( f(x) = 0 \).

Step 2: Method 1 - Factoring the Quadratic Equation

To find the intercepts using factoring, we attempt to express the quadratic \[ ax^2 + bx + c = 0 \] as a product of two binomials: \[ (px + q)(rx + s) = 0 \]. If factored, the horizontal intercepts are where \( px + q = 0 \) and \( rx + s = 0 \). Solving these equations gives the \( x \)-intercepts.

Step 3: Method 2 - Quadratic Formula

If the quadratic equation is not easily factorable, another method is using the quadratic formula:\[ x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a} \].This formula gives the solutions to the equation \( ax^2 + bx + c = 0 \), providing the horizontal intercepts of the function.

Key Concepts

Horizontal Intercepts

Horizontal intercepts, also known as x-intercepts, are the points where the graph of a quadratic function crosses the x-axis. This means that at these points, the value of the function, or y-value, is zero. To find these intercepts for a quadratic described by \[f(x) = ax^2 + bx + c \]where \( a eq 0 \), we need to solve the equation \( ax^2 + bx + c = 0 \). This leads us to two possibilities: either the equation factors neatly, or it doesn't and we need a different method. Let's dive deeper into these methods below.

Factoring Quadratics

Factoring quadratics is often the preferred method for solving simple quadratic equations, as it can be quicker and more intuitive. It involves expressing the quadratic equation \[ax^2 + bx + c = 0\]as a product of two binomials: \[(px + q)(rx + s) = 0.\]This step is easily possible when the numbers involved are such that they split nicely into two factors or roots.

• Check if the quadratic can be factored. Look for two numbers that multiply to \( ac \) (the product of \( a \) and \( c \)) and add up to \( b \).
• If you find these numbers, rewrite the middle term \( bx \) using them, making it possible to factor by grouping.
• Set each factor equal to zero, \( px + q = 0 \) and \( rx + s = 0 \), solve for \( x \) to get the intercepts.

If a quadratic cannot be easily factored using integers, you might have to draw on the quadratic formula as a more reliable approach.

Quadratic Formula

The quadratic formula is a powerful tool because it can find the horizontal intercepts of any quadratic function, regardless of whether it can be factored easily or not. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This gives us the solutions to the equation \( ax^2 + bx + c = 0 \), which are precisely the x-values where the graph intersects the x-axis.

• Start by identifying the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant, \( b^2 - 4ac \). This number tells you about the nature of the roots:
• If positive, there are two real and distinct intercepts.
• If zero, there is exactly one real intercept (a repeated root).
• If negative, there are no real intercepts; the solutions will be complex numbers.
• Substitute \( a \), \( b \), and \( c \) into the quadratic formula and solve for \( x \) to find the intercepts.

Whether factoring or using the quadratic formula, these methods are fundamental for analyzing quadratic functions and understanding the geometry of parabolas.