Question

select the correct answer.
what is the solution set of this system of equations?
y = -x² + 2x + 4
x + y = 4
a. (0, 4) and (4, 0)
b. (7, -3) and (0, 4)
c. (0, 4) and (3, 1)
d. (7, -3) and (4, 0)

Explanation:

Step1: Solve the linear equation for y

From \( x + y = 4 \), we can express \( y \) as \( y = 4 - x \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 4 - x \) into \( y = -x^2 + 2x + 4 \):

$$ 4 - x = -x^2 + 2x + 4 $$

Step3: Simplify the equation

Subtract 4 from both sides:

$$ -x = -x^2 + 2x $$

Add \( x^2 \) and subtract 2x from both sides:

$$ x^2 - 3x = 0 $$

Factor out x:

$$ x(x - 3) = 0 $$

Step4: Solve for x

Set each factor equal to zero:
\( x = 0 \) or \( x - 3 = 0 \) (so \( x = 3 \))

Step5: Find the corresponding y values

• When \( x = 0 \), substitute into \( y = 4 - x \): \( y = 4 - 0 = 4 \), so the point is \( (0, 4) \).
• When \( x = 3 \), substitute into \( y = 4 - x \): \( y = 4 - 3 = 1 \), so the point is \( (3, 1) \).

Answer:

C. \((0, 4)\) and \((3, 1)\)