https://file.aiquickdraw.com/mathbot/file/1725689140165impawhjd.jpg

The equation given is:

\[
y = ax^2 + bx + 1
\]

We are told that the curve has a gradient of 2 at the point \( (1, 0) \).

### Step-by-step solution:

1. **Differentiate the equation** to find the gradient (or slope) function:
   \[
   \frac{dy}{dx} = 2ax + b
   \]

2. **Use the point** \( (1, 0) \) to determine the gradient at \( x = 1 \). The gradient is given as 2, so:
   \[
   2a(1) + b = 2
   \]
   Simplifying:
   \[
   2a + b = 2 \quad \text{(Equation 1)}
   \]

3. **Substitute** \( (1, 0) \) into the original equation \( y = ax^2 + bx + 1 \) to get a second equation:
   \[
   0 = a(1)^2 + b(1) + 1
   \]
   This simplifies to:
   \[
   a + b + 1 = 0
   \]
   Rearranging:
   \[
   a + b = -1 \quad \text{(Equation 2)}
   \]

4. **Solve the system of equations**:
   From Equation 1:
   \[
   2a + b = 2
   \]
   From Equation 2:
   \[
   a + b = -1
   \]
   Subtract Equation 2 from Equation 1:
   \[
   (2a + b) - (a + b) = 2 - (-1)
   \]
   Simplifying:
   \[
   a = 3
   \]

5. **Substitute \( a = 3 \)** back into Equation 2:
   \[
   3 + b = -1
   \]
   Solving for \( b \):
   \[
   b = -4
   \]

### Final Answer:
The values of \( a \) and \( b \) are:
\[
a = 3, \quad b = -4
\]

Would you like more details or have any questions about this solution?

---

Here are five related questions to expand on this topic:

1. What is the significance of the gradient of a curve at a particular point?
2. How would the equation change if the gradient at \( x = 1 \) were different?
3. Can you find the equation of the tangent line to the curve at the point \( (1, 0) \)?
4. What happens to the shape of the parabola as the values of \( a \) and \( b \) change?
5. How would you verify that the values of \( a = 3 \) and \( b = -4 \) are correct?

**Tip**: When solving systems of equations, always double-check the substitutions to avoid calculation errors.

The curve with equation y = ax^2 + bx + 1 has a gradient of 2 at the point (1, 0).

Solve for the coefficients a and b of the quadratic equation y = ax^2 + bx + 1, given that the gradient at the point (1, 0) is 2. Learn step-by-step how to differentiate the equation and solve a system of linear equations. Suitable for 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