The curve y=ax square +bx passes through (2,0) and (4,8) Find a and b

We are given the curve equation \( y = ax^2 + bx \) and two points: \( (2, 0) \) and \( (4, 8) \). Using these points, we can set up a system of equations to solve for \( a \) and \( b \).

### Step 1: Use the point \( (2, 0) \)

Substitute \( x = 2 \) and \( y = 0 \) into the equation \( y = ax^2 + bx \):

\[
0 = a(2^2) + b(2)
\]
\[
0 = 4a + 2b
\]
\[
2a + b = 0 \quad \text{(Equation 1)}
\]

### Step 2: Use the point \( (4, 8) \)

Substitute \( x = 4 \) and \( y = 8 \) into the equation \( y = ax^2 + bx \):

\[
8 = a(4^2) + b(4)
\]
\[
8 = 16a + 4b
\]
\[
4a + b = 2 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations

From Equation 1: \( 2a + b = 0 \), we can express \( b \) in terms of \( a \):
\[
b = -2a
\]

Substitute this into Equation 2:
\[
4a + (-2a) = 2
\]
\[
2a = 2
\]
\[
a = 1
\]

Now, substitute \( a = 1 \) into \( b = -2a \):
\[
b = -2(1) = -2
\]

### Final Answer:
The values of \( a \) and \( b \) are:
\[
a = 1, \quad b = -2
\]

Would you like a further explanation of any of the steps or additional details?

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Here are 5 questions that expand on this concept:
1. How would the curve change if \( b = 0 \)?
2. Can we find the vertex of the parabola \( y = ax^2 + bx \)?
3. How do the signs of \( a \) and \( b \) affect the shape of the parabola?
4. What happens if the points \( (2, 0) \) and \( (4, 8) \) are switched?
5. How do you determine whether the curve opens upwards or downwards?

**Tip:** When solving systems of equations, always try to express one variable in terms of the other to simplify the substitution process.

The curve y=ax^2 + bx passes through (2,0) and (4,8). Find a and b.

Learn how to solve the quadratic equation y = ax^2 + bx that passes through points (2,0) and (4,8). This step-by-step guide explains how to set up and solve a system of equations to find the values of a and b. Suitable for students in Grades 9-11.

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