https://file.aiquickdraw.com/mathbot/file/172638426240594ybrqyj.jpg

The system of equations given is:

1. \( y = 2x^2 - 21x + 64 \)
2. \( y = 3x + a \)

We are asked to find the value of \( x \) at the intersection point of the two graphs, where the system has exactly one solution. This means that the quadratic equation must touch the line at one point, implying the quadratic has exactly one solution (this occurs when the discriminant of the quadratic is zero).

### Step 1: Set the equations equal to each other
Since both equations are equal to \( y \), we can set them equal:

\[
2x^2 - 21x + 64 = 3x + a
\]

### Step 2: Rearrange the equation
Move all terms to one side to form a quadratic equation:

\[
2x^2 - 21x - 3x + 64 - a = 0
\]

Simplify:

\[
2x^2 - 24x + (64 - a) = 0
\]

### Step 3: Apply the condition for one solution
For the quadratic equation to have exactly one solution, the discriminant must be zero. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is:

\[
\Delta = b^2 - 4ac
\]

In our case, \( a = 2 \), \( b = -24 \), and \( c = 64 - a \). Substituting into the discriminant formula:

\[
\Delta = (-24)^2 - 4(2)(64 - a) = 0
\]

Simplify:

\[
576 - 8(64 - a) = 0
\]

\[
576 - 512 + 8a = 0
\]

\[
64 + 8a = 0
\]

### Step 4: Solve for \( a \)
Solving the above equation for \( a \):

\[
8a = -64
\]

\[
a = -8
\]

### Step 5: Find the value of \( x \)
Now substitute \( a = -8 \) back into the linear equation \( y = 3x + a \):

\[
y = 3x - 8
\]

Substitute this into the quadratic equation:

\[
2x^2 - 21x + 64 = 3x - 8
\]

Simplifying:

\[
2x^2 - 24x + 72 = 0
\]

Divide by 2:

\[
x^2 - 12x + 36 = 0
\]

Factoring:

\[
(x - 6)^2 = 0
\]

So, \( x = 6 \).

### Final Answer:
The value of \( x \) is \( 6 \).

Would you like further details or have any questions? Here are some related questions:

1. What is the significance of the discriminant in quadratic equations?
2. How can we determine if two curves intersect at exactly one point?
3. What would happen if the discriminant were positive or negative?
4. How do you solve a system of equations involving both linear and quadratic functions?
5. Can you explain how to factor a quadratic equation like \( (x - 6)^2 = 0 \)?

**Tip:** Always check for conditions like "exactly one solution" when solving quadratic systems, as it often involves using the discriminant.

In the given system of equations, a is a constant. The graphs of the equations in the given system intersect at exactly one point, (x, y), in the xy-plane. What is the value of x?

Learn how to solve a system of equations where a quadratic and a linear equation intersect at exactly one point. This problem covers the discriminant method for determining when curves touch at one point. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation