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!