In the xy-coordinate plane, the graph of `y = 5x^(2)-12x` intersects the grphs of `y=-2` at points (0,0) and (a,b). What is the value of a ?

Explanation

To find the value of \( a \) where the graphs of \( y = 5x^2 - 12x \) and \( y = -2 \) intersect, we can follow these steps:

### Step 1: Set the equations equal to each other
We start by setting the two equations equal to each other since at the points of intersection, the \( y \)-values will be the same.

\[ 5x^2 - 12x = -2 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to set it to zero:

\[ 5x^2 - 12x + 2 = 0 \]

### Step 3: Use the quadratic formula
Now, we will use the quadratic formula to solve for \( x \). The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, \( a = 5 \), \( b = -12 \), and \( c = 2 \).

### Step 4: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = (-12)^2 - 4 \cdot 5 \cdot 2 = 144 - 40 = 104 \]

### Step 5: Substitute into the quadratic formula
Now we can substitute the values into the quadratic formula:

\[ x = \frac{-(-12) \pm \sqrt{104}}{2 \cdot 5} = \frac{12 \pm \sqrt{104}}{10} \]

### Step 6: Simplify \(\sqrt{104}\)
We can simplify \(\sqrt{104}\):

\[ \sqrt{104} = \sqrt{4 \cdot 26} = 2\sqrt{26} \]

### Step 7: Substitute back to find \( x \)
Now substituting back, we have:

\[ x = \frac{12 \pm 2\sqrt{26}}{10} = \frac{6 \pm \sqrt{26}}{5} \]

### Step 8: Identify the points of intersection
The two values of \( x \) are:

\[ x_1 = \frac{6 + \sqrt{26}}{5} \quad \text{and} \quad x_2 = \frac{6 - \sqrt{26}}{5} \]

Since we know one intersection point is \( (0, 0) \), we need to find the other intersection point, which corresponds to \( a \).

### Step 9: Determine the value of \( a \)
The value of \( a \) is the \( x \)-coordinate of the other intersection point. We can choose either \( x_1 \) or \( x_2 \). Since the problem states that the points of intersection are \( (0,0) \) and \( (a,b) \), we can take:

\[ a = \frac{6 - \sqrt{26}}{5} \]

### Final Answer
Thus, the value of \( a \) is:

\[ a = \frac{6 - \sqrt{26}}{5} \]

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