Final answer:
To solve the given system consisting of a quadratic equation (y = x² - 3x - 5) and a linear equation (y = x + 3), we set them equal to each other, rearrange the terms, and solve for x using the quadratic formula, obtaining x = 2 ± √3. We then use these x values to find corresponding y values.
Step-by-step explanation:
To solve the system of quadratic and linear equations:
- Quadratic Equation: y = x² - 3x - 5
- Linear Equation: y = x + 3
We set these equations equal to each other because they both equal y:
x + 3 = x² - 3x - 5
Rearrange the equation to find x:
0 = x² - 3x - 5 - (x + 3)
0 = x² - 4x - 8
Solve for x using the quadratic formula:
x = ± √(b² - 4ac) / (2a), where a = 1, b = -4, c = -8
Plugging in these values, we get two solutions for x:
x = (4 ± √(16 + 32)) / 2
x = (4 ± √48) / 2
x = (4 ± 4√3) / 2
Which simplifies to:
x = 2 ± √3
Substitute these x values back into either the quadratic or linear equation to find corresponding y values.