Find out the points where the curve y=4x2+x*y*=4*x*2+*x* and the straight line y=2x−3*y*=2*x*−3 intersect with each other.

(32,0)(23​,0) and (32,212)(23​,221​).

Only at the origin.

The curve and the straight line do not intersect.

(1,−1)(1,−1) and (1,5)(1,5).

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 (\( < 0 \)).

Find out the points where the curve y = 4x^2 + x and the straight line y = 2x − 3 intersect with each other.

Discover how to find the points of intersection between the quadratic curve y = 4x^2 + x and the straight line y = 2x - 3. This problem involves solving a quadratic equation and analyzing the discriminant to determine if real solutions exist. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation