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.