You are given two equations:

1. \( y = 5x^2 \)
2. \( y = x + 6 \)

To find the points where these two curves intersect, we need to set the two equations equal to each other since both are equal to \( y \):

\[
5x^2 = x + 6
\]

Now, rearrange the equation:

\[
5x^2 - x - 6 = 0
\]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a = 5 \), \( b = -1 \), and \( c = -6 \). We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values of \( a \), \( b \), and \( c \) into the formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(5)(-6)}}{2(5)}
\]

Simplify:

\[
x = \frac{1 \pm \sqrt{1 + 120}}{10}
\]
\[
x = \frac{1 \pm \sqrt{121}}{10}
\]
\[
x = \frac{1 \pm 11}{10}
\]

Now, solve for the two possible values of \( x \):

1. \( x = \frac{1 + 11}{10} = \frac{12}{10} = 1.2 \)
2. \( x = \frac{1 - 11}{10} = \frac{-10}{10} = -1 \)

Thus, the solutions for \( x \) are \( x = 1.2 \) and \( x = -1 \).

To find the corresponding \( y \)-values, substitute these \( x \)-values back into either of the original equations. I'll use \( y = x + 6 \):

For \( x = 1.2 \):
\[
y = 1.2 + 6 = 7.2
\]

For \( x = -1 \):
\[
y = -1 + 6 = 5
\]

Therefore, the points of intersection are:

- \( (1.2, 7.2) \)
- \( (-1, 5) \)

Would you like any further details or clarifications?

Here are five related questions:
1. How can we solve quadratic equations graphically?
2. What are the different methods for solving quadratic equations?
3. Can a quadratic and a linear equation intersect at more than two points?
4. How do we interpret the discriminant of a quadratic equation geometrically?
5. What is the significance of solving systems of equations in real-world applications?

**Tip:** When solving quadratic equations, always check whether factoring is possible before using the quadratic formula—it can simplify your work!

Solve the system of equations: y = 5x² and y = x + 6.

Discover how to solve the system of equations y = 5x² and y = x + 6 using algebraic methods. This step-by-step solution walks through solving a quadratic and linear equation system, applying the quadratic formula. Ideal for Grades 8-10 students.

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