how to find out if a 2 quadratic equations are intersecting at some point

To find out if two quadratic equations intersect at some point, you need to determine if there is a common solution to both equations. This involves solving the system of quadratic equations. Here's a step-by-step approach:

1. **Write down the two quadratic equations** in the standard form:
   \[ y = ax^2 + bx + c \]
   \[ y = dx^2 + ex + f \]

2. **Set the equations equal to each other** to find the x-values where they might intersect:
   \[ ax^2 + bx + c = dx^2 + ex + f \]

3. **Combine like terms** to form a single quadratic equation:
   \[ (a-d)x^2 + (b-e)x + (c-f) = 0 \]

4. **Solve the resulting quadratic equation** for x using the quadratic formula:
   \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
   where \( A = a - d \), \( B = b - e \), and \( C = c - f \).

5. **Determine the nature of the roots** by evaluating the discriminant \( \Delta = B^2 - 4AC \):
   - If \( \Delta > 0 \), there are two distinct solutions (intersections).
   - If \( \Delta = 0 \), there is one solution (tangency).
   - If \( \Delta < 0 \), there are no real solutions (no intersection).

6. **Verify the y-values** for the x-values obtained to confirm the intersection points:
   - Substitute the x-values back into either original quadratic equation to find the corresponding y-values.
   - Ensure that the y-values match for both equations at these x-values.

### Example
Consider the equations:
\[ y = x^2 + 2x + 1 \]
\[ y = -x^2 + 4x + 3 \]

1. Set them equal:
   \[ x^2 + 2x + 1 = -x^2 + 4x + 3 \]

2. Combine like terms:
   \[ 2x^2 - 2x - 2 = 0 \]
   \[ x^2 - x - 1 = 0 \]

3. Solve for x using the quadratic formula:
   \[ x = \frac{1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
   \[ x = \frac{1 \pm \sqrt{1 + 4}}{2} \]
   \[ x = \frac{1 \pm \sqrt{5}}{2} \]

4. Verify the y-values by substituting these x-values back into either original equation.

Would you like to see the detailed calculations for these intersections, or do you have any other questions?

Here are 5 numbered questions you might want to ask next:
1. How to solve a quadratic equation using the quadratic formula?
2. How to find the discriminant of a quadratic equation?
3. How to graph two quadratic functions to visualize their intersection?
4. How to determine if a quadratic equation has real solutions?
5. How to solve a system of linear equations?

**Tip:** Always verify your solutions by substituting them back into the original equations to ensure they satisfy both equations.

Learn how to determine if two quadratic equations intersect at some point using the quadratic formula. This detailed guide covers setting up and solving the system of equations, calculating the discriminant, and verifying intersection points. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation