To find the points where the curve and the straight line intersect, we need to solve their equations simultaneously. The given equations are:

1. Curve: $y = 4x^2 + x$
2. Straight line: $y = 2x - 3$

Step 1: Set the equations equal

At the points of intersection, the $y$-coordinates will be the same, so we set the right-hand sides of both equations equal to each other:

$4x^2 + x = 2x - 3$

Step 2: Rearrange the equation

Move all terms to one side of the equation:

$4x^2 + x - 2x + 3 = 0$ $4x^2 - x + 3 = 0$

Step 3: Solve the quadratic equation

We now solve the quadratic equation $4x^2 - x + 3 = 0$. The general solution to a quadratic equation $ax^2 + bx + c = 0$ is given by:

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

For $4x^2 - x + 3 = 0$, the coefficients are:

• $a = 4$
• $b = -1$
• $c = 3$

Plugging these values into the quadratic formula:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(3)}}{2(4)}$ $x = \frac{1 \pm \sqrt{1 - 48}}{8}$ $x = \frac{1 \pm \sqrt{-47}}{8}$

Since $\sqrt{-47}$ is an imaginary number, there are no real solutions to this equation.

Conclusion:

Since the quadratic equation has no real solutions, the curve and the straight line do not intersect.

Would you like more details on how to handle complex roots, or have any further questions?

Here are 5 related questions you may find helpful:

1. How do you solve quadratic equations when there are real solutions?
2. How do you find the points of intersection between a parabola and a line geometrically?
3. What are complex numbers, and how do they apply to quadratic equations?
4. Can two curves in the plane never intersect? How would you determine that in general?
5. What is the discriminant, and how does it tell us the nature of the roots of a quadratic equation?

Tip: The discriminant $b^2 - 4ac$ in a quadratic equation tells us if the solutions are real ($\geq 0$) or complex ($< 0$).