In the xy-plane, the graph of `y + 3x =5x ^(2) +6 and y -6 =2x` intersect at points (0,6) and (a,b). What is the value of b ?

Explanation

To find the value of \( b \) where the graphs of the equations intersect, we will follow these steps:

1. **Write down the equations**:
The first equation is given as:
\[ y + 3x = 5x^2 + 6 \]
The second equation is given as:
\[ y - 6 = 2x \]

2. **Rearrange the second equation to express \( y \)**:
From the second equation, we can isolate \( y \):
\[ y = 2x + 6 \]

3. **Substitute \( y \) in the first equation**:
Now, we substitute \( y \) from the second equation into the first equation:
\[ (2x + 6) + 3x = 5x^2 + 6 \]

4. **Combine like terms**:
Simplifying the left side:
\[ 2x + 6 + 3x = 5x^2 + 6 \]
This simplifies to:
\[ 5x + 6 = 5x^2 + 6 \]

5. **Move all terms to one side**:
Subtract \( 5x + 6 \) from both sides:
\[ 5x^2 + 6 - 5x - 6 = 0 \]
This simplifies to:
\[ 5x^2 - 5x = 0 \]

6. **Factor the equation**:
Factor out the common term:
\[ 5x(x - 1) = 0 \]

7. **Set each factor to zero**:
This gives us two possible solutions:
\[ 5x = 0 \quad \text{or} \quad x - 1 = 0 \]
Thus, we have:
\[ x = 0 \quad \text{or} \quad x = 1 \]

8. **Find corresponding \( y \) values**:
We will now find the \( y \) values for both \( x \) values using the equation \( y = 2x + 6 \):
- For \( x = 0 \):
\[ y = 2(0) + 6 = 6 \]
- For \( x = 1 \):
\[ y = 2(1) + 6 = 8 \]

9. **Identify the intersection points**:
The points of intersection are:
- \( (0, 6) \)
- \( (1, 8) \)

10. **Determine the value of \( b \)**:
From the intersection point \( (1, 8) \), we see that \( b = 8 \).

Thus, the value of \( b \) is:
\[ \boxed{8} \]