Problem 54 - Algebra for College Students

Question

Give a step-by-step explanation of how to find the \(x\) intercepts of the graph of the function \(f(x)=2 x^{2}+7 x-4\).

Step-by-Step Solution

Answer

The \(x\)-intercepts are \(x = \frac{1}{2}\) and \(x = -4\).

Step 1: Identify the Function

The given function is a quadratic function: \( f(x) = 2x^2 + 7x - 4 \). Our task is to find the \(x\)-intercepts of this function.

Step 2: Set Function Equal to Zero

To find the \(x\)-intercepts, we need to solve for \(x\) when \(f(x) = 0\). So, we set the equation \(2x^2 + 7x - 4 = 0\).

Step 3: Apply the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 7 \), and \( c = -4 \). We will use this to find the values of \(x\).

Step 4: Calculate the Discriminant

Compute the discriminant \( b^2 - 4ac \). Substituting the values, we get \( 7^2 - 4(2)(-4) = 49 + 32 = 81 \).

Step 5: Compute the Roots

Using the quadratic formula with the discriminant, we find the roots: \[ x = \frac{-7 \pm \sqrt{81}}{4} \]. Calculate separately for the plus and minus cases.

Step 6: Simplify the Roots

Calculate \( x = \frac{-7 + 9}{4} = \frac{2}{4} = \frac{1}{2} \) and \( x = \frac{-7 - 9}{4} = \frac{-16}{4} = -4 \). These are the \(x\)-intercepts.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, meaning that the highest power of the variable is squared. In general, it can be expressed in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
Quadratic functions create a parabolic curve when graphed on a coordinate plane. The shape of the parabola depends on the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Understanding quadratic functions is fundamental to solving for \( x \)-intercepts, finding the vertex, and analyzing the symmetry of the graph.
When we solve for \( x \) in a quadratic function, we're typically looking for the points where the curve intersects the \( x \)-axis. These are the solutions or "roots" of the quadratic equation \( ax^2 + bx + c = 0 \).

Quadratic Formula

The quadratic formula provides a straightforward method for finding the \( x \)-intercepts of a quadratic function. If you have a quadratic equation in the form \( ax^2 + bx + c = 0 \), the solutions for \( x \) are given by the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows you to calculate the precise values of \( x \) that satisfy the equation.
It works universally for all quadratics, as long as \( a \), \( b \), and \( c \) are real numbers. To apply it, you simply substitute these coefficients into the formula:

• Calculate \(-b\).
• Find the square root of the discriminant \(b^2 - 4ac\).
• Divide by \(2a\).

The "plus or minus" symbol (\(\pm\)) indicates that you'll find two solutions, which represent the two possible values at which the parabola intersects the \( x \)-axis.

Discriminant

The discriminant is a crucial component of the quadratic formula nestled under the square root symbol. It is represented as \( b^2 - 4ac \) and determines the nature of the roots of the quadratic equation.
By examining the discriminant, you can predict the number and type of roots without solving the entire equation. Here's how it works:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots. The parabola intersects the \( x \)-axis at two points.
• If \( b^2 - 4ac = 0 \), there is exactly one real root (or a repeated root). The parabola touches the \( x \)-axis at a single point (the vertex).
• If \( b^2 - 4ac < 0 \), there are no real roots. The parabola does not intersect the \( x \)-axis, indicating the roots are complex or imaginary numbers.

By first calculating the discriminant, you can quickly assess the potential solutions to the quadratic equation, which is particularly useful in many areas of mathematics and applied sciences.