What is the solution set for the system of equations defined by y = x^2 – 3x + 4 and x + y = 8?

To find the solution set of the system of equations given by:

• Equation 1: y = x² – 3x + 4
• Equation 2: x + y = 8

We start by substituting Equation 1 into Equation 2. This means we will replace y in Equation 2 with the expression from Equation 1.

1. Substitute y in Equation 2:

x + (x2 - 3x + 4) = 8

2. Simplify the equation:

x + x2 - 3x + 4 = 8

x2 - 2x + 4 - 8 = 0

x2 - 2x - 4 = 0

3. Now, we will apply the quadratic formula to solve for x. The quadratic formula is:

x = (-b ± √(b² – 4ac)) / (2a)

For our equation:

• a = 1
• b = -2
• c = -4

4. Calculate the discriminant:

b2 - 4ac = (-2)2 - 4 * 1 * (-4) = 4 + 16 = 20

5. Now plug these values into the quadratic formula:

x = (2 ± √20) / 2

x = (2 ± 2√5) / 2

x = 1 ± √5

So, we find two possible values for x:

• x = 1 + √5
• x = 1 – √5

6. Substitute back to find y for each value of x:

• For x = 1 + √5:

y = (1 + √5)2 - 3(1 + √5) + 4

y = (1 + 2√5 + 5) - 3 - 3√5 + 4 = 3 - √5

• For x = 1 – √5:

y = (1 - √5)2 - 3(1 - √5) + 4

y = (1 - 2√5 + 5) - 3 + 3√5 + 4 = 3 + √5

Thus, the solution set for the system of equations is:

• Solution 1: (1 + √5, 3 – √5)
• Solution 2: (1 – √5, 3 + √5)

In conclusion, the solution set of the given system of equations comprises two points, which can be summarized as:

{ (1 + √5, 3 - √5), (1 - √5, 3 + √5) }