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The image contains a problem that involves finding the roots of a quadratic equation and plotting them on a graph. Let's break down and solve the problem step by step.

### Problem Details:
1. The quadratic equation given is:
   \[
   -4t^2 + t - 6 = 0
   \]
2. The instructions seem to ask for the roots of the equation and plotting these roots on the provided graph.

### Step 1: Solve the Quadratic Equation
The general form of a quadratic equation is:
\[
at^2 + bt + c = 0
\]
For the given equation:
\[
-4t^2 + t - 6 = 0
\]
Here, \(a = -4\), \(b = 1\), and \(c = -6\).

To find the roots, we can use the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Plugging in the values:
\[
t = \frac{-1 \pm \sqrt{1^2 - 4(-4)(-6)}}{2(-4)}
\]
\[
t = \frac{-1 \pm \sqrt{1 - 96}}{-8}
\]
\[
t = \frac{-1 \pm \sqrt{-95}}{-8}
\]

### Step 2: Analyze the Discriminant
The discriminant \( \Delta = b^2 - 4ac \) is negative (\(\Delta = -95\)), which means the equation has **no real roots** but rather **complex roots**.

### Step 3: Plotting on the Graph
Since the roots are complex, they cannot be plotted on a real number line, which is what the grid provided seems to represent.

### Conclusion:
The equation does not have real roots, so nothing should be plotted on the graph for real-number roots.

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Would you like any further details on how to solve quadratic equations with complex roots, or have any questions about this problem? Here are five related questions:

1. How do you find the roots of a quadratic equation with a negative discriminant?
2. What do complex roots represent in a quadratic equation?
3. How can the quadratic formula be derived from the standard form

-4t^2 + t - 6 = 0. Find the roots and plot them on the graph.

Learn how to solve the quadratic equation -4t^2 + t - 6 = 0 and analyze the roots. This detailed solution covers key algebraic concepts such as using the quadratic formula and understanding the discriminant. Ideal for students in Grades 9-12.

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