https://file.aiquickdraw.com/mathbot/file/1726620425488ehbq6xv8.jpg

The system of equations provided in the image is:

\[
\begin{cases}
y = x^2 + 2x + 5 \\
y = 5x + 15
\end{cases}
\]

To solve for \(x\) and \(y\), we can set the two expressions for \(y\) equal to each other since both are equal to \(y\).

\[
x^2 + 2x + 5 = 5x + 15
\]

Now, let's solve for \(x\).

First, move all terms to one side of the equation:

\[
x^2 + 2x + 5 - 5x - 15 = 0
\]

Simplifying:

\[
x^2 - 3x - 10 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}
\]

\[
x = \frac{3 \pm \sqrt{9 + 40}}{2}
\]

\[
x = \frac{3 \pm \sqrt{49}}{2}
\]

\[
x = \frac{3 \pm 7}{2}
\]

Thus, we get two possible values for \(x\):

\[
x = \frac{3 + 7}{2} = 5 \quad \text{or} \quad x = \frac{3 - 7}{2} = -2
\]

Now, substitute these values of \(x\) into either equation to find the corresponding values of \(y\).

1. When \(x = 5\):
\[
y = 5(5) + 15 = 25 + 15 = 40
\]

2. When \(x = -2\):
\[
y = 5(-2) + 15 = -10 + 15 = 5
\]

So, the solutions are:

\[
(5, 40) \quad \text{and} \quad (-2, 5)
\]

Would you like any further details or explanations?

Here are 5 related questions you could explore:
1. What are the roots of the quadratic equation \(x^2 - 3x - 10 = 0\)?
2. How do you verify the solutions of a system of equations?
3. What is the significance of solving quadratic equations in algebra?
4. Can you solve this system of equations graphically?
5. How can the quadratic formula be derived from completing the square?

**Tip:** Always check your solutions by substituting them back into the original system to verify they satisfy both equations.

Solve the system of equations: y = x^2 + 2x + 5 and y = 5x + 15.

Learn how to solve a system of equations involving a quadratic and linear equation: y = x^2 + 2x + 5 and y = 5x + 15. This step-by-step solution covers the quadratic formula and solving for both x and y. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation