# Which Parabola Will Have One Real Solution With the Line Y=X-5

Which Parabola Will Have One Real Solution With the Line Y=X-5?

Introduction:

When studying quadratic equations, it is often important to understand the relationship between a parabola and a given line. In this article, we will explore the scenario in which a parabola intersects a line at only one point, resulting in a single real solution. Specifically, we will focus on the line y = x – 5 and analyze which parabola will exhibit this characteristic. Additionally, we will address frequently asked questions related to this topic.

Understanding the Parabola:

A parabola is a U-shaped curve that can open upwards or downwards. It is defined by its vertex, which is the highest or lowest point on the curve, and its axis of symmetry, which is a vertical line passing through the vertex. The general equation of a parabola can be written as y = ax^2 + bx + c, where a, b, and c are constants.

Intersection of a Parabola and a Line:

When a parabola intersects a line, the point(s) of intersection represent the solutions to the system of equations formed by equating the equations of the parabola and the line. In the case of the line y = x – 5, this equation can be substituted into the general equation of a parabola to find the values of x where the two curves intersect.

To determine which parabola will have one real solution with the line y = x – 5, we need to find the discriminant of the quadratic equation. The discriminant, denoted by Δ, is calculated as b^2 – 4ac. It helps determine the number and nature of the solutions to a quadratic equation.

Analyzing the Discriminant:

For a parabola to have one real solution with the line y = x – 5, the discriminant must be equal to zero. This condition arises when the quadratic equation has one real root, indicating that the parabola and the line have exactly one point of intersection.

Let’s consider the quadratic equation y = ax^2 + bx + c and substitute y = x – 5 into it:

x – 5 = ax^2 + bx + c

Rearranging the terms, we obtain the quadratic equation:

ax^2 + (b – 1)x + (c + 5) = 0

To find the discriminant, we substitute the coefficients into the formula:

Δ = (b – 1)^2 – 4ac

For the discriminant to be zero, we have:

(b – 1)^2 – 4ac = 0

Simplifying this equation, we get:

b^2 – 2b + 1 – 4ac = 0

b^2 – 2b + 1 = 4ac

Now, let’s examine a frequently asked question related to this topic.

FAQs:

Q1: What does it mean for a parabola to have one real solution with a given line?

A1: When a parabola intersects a line at only one point, it means that the two curves have exactly one common point of intersection. In other words, the quadratic equation derived from the parabola and the equation of the line will have only one real root.

Conclusion:

Determining which parabola will have one real solution with the line y = x – 5 requires analyzing the discriminant of the quadratic equation. By equating the given line equation with the general equation of a parabola, we can derive a quadratic equation and evaluate its discriminant. When the discriminant is zero, the parabola and the line intersect at a single point. Understanding the relationship between parabolas and lines is crucial in various mathematical applications and problem-solving scenarios.