Problem 54 - Introductory Algebra for College Students

Question

If a parabola has two \(x\) -intercepts, explain how to find them.

Step-by-Step Solution

Answer

The x-intercepts of a parabola can be found by first identifying the coefficients of the quadratic equation, then substituting these into the quadratic formula. By calculating the discriminant, we can find out how many x-intercepts there are. If the discriminant is positive, we can solve for the two x-intercepts.

Step 1: Identify the coefficients of the quadratic equation

If your quadratic equation is written in the form \( ax^2 + bx + c = 0 \), identify the coefficients \( a \), \( b \), and \( c \). These will be used in the quadratic formula.

Step 2: Apply the quadratic formula

Substitute the coefficients found in Step 1 into the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This will give you a formula for the x-intercepts or roots of the equation.

Step 3: Calculate the discriminant

Calculate the value of the discriminant, which is the part under the square root in the quadratic formula, \( b^2 - 4ac \). If this value is positive, there will be 2 x-intercepts, if it's 0 there will be only one x-intercept, and if it's negative there are no x-intercepts.

Step 4: Find the x-intercepts

If the discriminant is positive, solve the quadratic formula for \( x \) to find the two x-intercepts. If the discriminant is 0, there is only 1 x-intercept, and if it's negative, there are no x-intercepts.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions, or roots, of quadratic equations. A quadratic equation has the form \( ax^2 + bx + c = 0 \), where \( a \) is not equal to zero. The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). To use the formula, one simply substitutes the values of \( a \) (coefficient of \(x^2\)), \( b \) (coefficient of \(x\)), and \( c \) (constant term) from the quadratic equation into the formula.

The symbols \pm indicate that the formula will provide two values for \( x \), corresponding to the two possible solutions to the equation. These solutions can be real or complex numbers depending on the value of the discriminant, which we'll discuss in the next section.

Discriminant of a Quadratic Equation

The discriminant is the part of the quadratic formula that resides under the square root symbol: \( b^2 - 4ac \). Knowing the value of the discriminant provides insight into the nature of the x-intercepts of the quadratic equation.

• If the discriminant is positive (greater than 0), there are two distinct real x-intercepts.
• If the discriminant is zero, there is exactly one real x-intercept, implying that the parabola touches the x-axis at one point.
• If the discriminant is negative (less than 0), there are no real x-intercepts, indicating that the parabola does not cross the x-axis, and the roots of the equation are complex numbers.

Understanding the discriminant can immediately tell us about the number of solutions and whether those solutions are real or complex without actually computing the roots.

Parabola

The parabola is the graph of a quadratic equation and is a U-shaped curve that can open upwards or downwards. The direction depends on the sign of the coefficient \( a \) in the equation \( ax^2 + bx + c = 0 \). If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.

A parabola has several key features, including the vertex, the axis of symmetry, and the x-intercepts (if they exist). The vertex is the highest or lowest point on the graph, and the axis of symmetry is a vertical line that passes through the vertex dividing the parabola into two mirrored halves. The x-intercepts are the points where the parabola crosses the x-axis and represent the solutions to the quadratic equation.

X-axis Intersections

Finding the x-axis intersections, or x-intercepts, of a parabola involves finding the values of \( x \) for which the quadratic equation \( ax^2 + bx + c = 0 \) yields a zero value of \( y \). These points indicate where the parabola touches or cuts through the x-axis.

To locate these intercepts, we use the quadratic formula, as already detailed. By interpreting the discriminant, we know prior to calculation if such points exist, and if the discriminant is positive, we can calculate and find the two points of intersection. These x-intercepts are of utmost importance as they provide the roots of the equation and are key elements in the graph's overall structure.