To solve the problem, we need to find the equation of the straight line PQ that intersects the graph of the function \( y = x^2 + 4 \) at the points where the quadratic equation \( x^2 - 3x + 2 = 0 \) has its roots. Let's break this down step-by-step. ### Step 1: Solve the Quadratic Equation The first step is to solve the quadratic equation \( x^2 - 3x + 2 = 0 \). We can factor this equation: \[ x^2 - 3x + 2 = (x - 1)(x - 2) = 0 \] Setting each factor to zero gives us the roots: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] So, the roots of the equation are \( x = 1 \) and \( x = 2 \). ### Step 2: Find the Corresponding y-values Next, we need to find the corresponding y-values for these x-values using the function \( y = x^2 + 4 \). 1. For \( x = 1 \): \[ y = 1^2 + 4 = 1 + 4 = 5 \] So, the point is \( (1, 5) \). 2. For \( x = 2 \): \[ y = 2^2 + 4 = 4 + 4 = 8 \] So, the point is \( (2, 8) \). ### Step 3: Determine the Slope of Line PQ Now we have two points on the line PQ: \( (1, 5) \) and \( (2, 8) \). We can find the slope \( m \) of the line using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points \( (1, 5) \) and \( (2, 8) \): \[ m = \frac{8 - 5}{2 - 1} = \frac{3}{1} = 3 \] ### Step 4: Use Point-Slope Form to Find the Equation of the Line Now that we have the slope, we can use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] Using the point \( (1, 5) \) and the slope \( m = 3 \): \[ y - 5 = 3(x - 1) \] Expanding this: \[ y - 5 = 3x - 3 \] \[ y = 3x + 2 \] ### Conclusion: The Equation of Line PQ Thus, the equation of the line PQ is: \[ y = 3x + 2 \] ### Step 5: Evaluate the Options Now, let's evaluate the options provided: - **A. \( y = 3x - 2 \)**: This is incorrect because it does not match our derived equation. - **B. \( y = 3x + 2 \)**: This is correct as it matches our derived equation. - **C. \( y = 3x - 4 \)**: This is incorrect because it does not match our derived equation. - **D. \( y = 3x + 4 \)**: This is incorrect because it does not match our derived equation. ### Revision Summary - The roots of the quadratic equation \( x^2 - 3x + 2 = 0 \) are \( x = 1 \) and \( x = 2 \). - The corresponding y-values from the function \( y = x^2 + 4 \) are \( (1, 5) \) and \( (2, 8) \). - The slope of the line connecting these points is \( 3 \), leading to the equation \( y = 3x + 2 \). - The correct answer is option **B**: \( y = 3x + 2 \).