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.

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