Find All Possible Values Of { X $}$ Such That The Given Line Is Tangent To The Graph Of { Y = F(x) $} . G I V E N : .Given: . G I V E N : { Y = Ax - 5 \} $ F(x) = 3x^2 - 4x - 2 }$ Example Answer Format { X = 3; X = 5 $ $

Mar 11, 2025 by ADMIN 222 views

**Tangent Line and Quadratic Function: Finding the Common Values**

Introduction

In mathematics, the concept of a tangent line is crucial in understanding the behavior of functions, particularly quadratic functions. A tangent line is a line that touches a curve at a single point, and in this case, we are interested in finding the values of x such that the given line is tangent to the graph of a quadratic function. In this article, we will explore the problem of finding all possible values of x that satisfy this condition.

Understanding the Problem

Given the line equation ${$ y = ax - 5 $}$ and the quadratic function ${$ f(x) = 3x^2 - 4x - 2 $}$, we want to find the values of x such that the line is tangent to the graph of the quadratic function. This means that the line touches the curve at a single point, and we need to find the x-coordinates of these points.

The Condition for Tangency

For the line to be tangent to the graph of the quadratic function, the equation ${$ f(x) = ax - 5 $}$ must have exactly one solution. This means that the quadratic equation ${$ 3x^2 - 4x - 2 = ax - 5 $}$ must have a discriminant of zero, since a quadratic equation with a discriminant of zero has exactly one solution.

Simplifying the Quadratic Equation

We can simplify the quadratic equation by combining like terms: ${$ 3x^2 - 4x - 2 = ax - 5 $}$ becomes ${$ 3x^2 - (4 + a)x + 3 = 0 $}$. This is a quadratic equation in the form ${$ ax^2 + bx + c = 0 $}$, where ${$ a = 3 $}$, ${$ b = -(4 + a) $}$, and ${$ c = 3 $}$.

The Discriminant

The discriminant of a quadratic equation is given by the formula ${$ D = b^2 - 4ac $}$. In this case, the discriminant is ${$ D = [-(4 + a)]^2 - 4(3)(3) $}$. We want to find the values of x such that the discriminant is zero, so we set ${$ D = 0 $}$ and solve for x.

Solving for x

Setting the discriminant equal to zero, we get ${$ [-(4 + a)]^2 - 4(3)(3) = 0 $}$. Expanding and simplifying, we get ${$ (4 + a)^2 - 36 = 0 $}$. This is a quadratic equation in a, and we can solve for a by factoring: ${$ (4 + a - 6)(4 + a + 6) = 0 $}$. This gives us two possible values for a: ${$ a = 2 $}$ and ${$ a = -10 $}$.

Finding the Values of x

Now that we have the values of a, we can substitute them back into the original equation ${$ 3x^2 - (4 + a)x + 3 = 0 $}$ and solve for x. For ${$ a = 2 $}$, we get ${$ 3x^2 - 6x + 3 = 0 $}$. This is a quadratic equation, and we can solve for x by factoring: ${$ (x - 1)(3x - 3) = 0 $}$. This gives us two possible values for x: ${$ x = 1 $}$ and ${$ x = 1 $}$. However, since the line is tangent to the graph of the quadratic function, we know that the line touches the curve at a single point, so we must have ${$ x = 1 $}$.

For ${$ a = -10 $}$, we get ${$ 3x^2 + 6x + 3 = 0 $}$. This is a quadratic equation, and we can solve for x by factoring: ${$ (3x + 1)(x + 3) = 0 $}$. This gives us two possible values for x: ${$ x = -1/3 $}$ and ${$ x = -3 $}$. However, since the line is tangent to the graph of the quadratic function, we know that the line touches the curve at a single point, so we must have ${$ x = -1/3 $}$.

Conclusion

In this article, we have found the values of x such that the given line is tangent to the graph of the quadratic function. We have used the condition for tangency, which is that the quadratic equation must have a discriminant of zero, and we have solved for x by substituting the values of a back into the original equation. We have found that the values of x are ${$ x = 1 $}$ and ${$ x = -1/3 $}$.

Example Answer Format

The values of x that satisfy the condition for tangency are ${$ x = 1; x = -1/3 $}$.

Discussion Category

This problem is a classic example of a tangent line and quadratic function problem, and it requires a deep understanding of the concepts of tangency and quadratic equations. The solution involves using the condition for tangency, which is that the quadratic equation must have a discriminant of zero, and solving for x by substituting the values of a back into the original equation. This problem is a great example of how to apply mathematical concepts to real-world problems.

Introduction

In our previous article, we explored the problem of finding the values of x such that the given line is tangent to the graph of a quadratic function. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the condition for tangency?

A: The condition for tangency is that the quadratic equation must have a discriminant of zero. This means that the quadratic equation ${$ ax^2 + bx + c = 0 $}$ must satisfy the condition ${$ b^2 - 4ac = 0 $}$.

Q: How do I find the values of x that satisfy the condition for tangency?

A: To find the values of x that satisfy the condition for tangency, you need to substitute the values of a back into the original equation and solve for x. This will give you the x-coordinates of the points where the line is tangent to the graph of the quadratic function.

Q: What if the quadratic equation has a discriminant of zero, but the line is not tangent to the graph of the quadratic function?

A: If the quadratic equation has a discriminant of zero, but the line is not tangent to the graph of the quadratic function, then the line is actually a secant line. This means that the line intersects the graph of the quadratic function at two or more points.

Q: Can the line be tangent to the graph of the quadratic function at more than one point?

A: No, the line cannot be tangent to the graph of the quadratic function at more than one point. If the line is tangent to the graph of the quadratic function at more than one point, then it is actually a secant line.

Q: How do I know if the line is tangent to the graph of the quadratic function?

A: To determine if the line is tangent to the graph of the quadratic function, you need to check if the quadratic equation has a discriminant of zero. If the discriminant is zero, then the line is tangent to the graph of the quadratic function.

Q: Can the line be tangent to the graph of the quadratic function if the quadratic equation has a discriminant of zero?

A: Yes, the line can be tangent to the graph of the quadratic function if the quadratic equation has a discriminant of zero. This is the condition for tangency.