Digital SAT Math Practice Question #1181

⚡ Hack the Test: For a parabola \( y = ax^2 + bx + d \) and a line \( y = mx + c \), the x-coordinate of the midpoint of their intersection points is given by the formula \( x_{mid} = \frac{-(b+m)}{2a} \). This formula is derived directly from Vieta's formulas. In this problem: Parabola: \( y = x^2 - 6x + 10 \), so \( a=1 \) and \( b=-6 \). Line: \( y = mx + c \). Midpoint x-coordinate: \( x_{mid} = 4 \). Substitute these values into the formula: \( 4 = \frac{-(-6 + m)}{2(1)} \) \( 4 = \frac{6 - m}{2} \) <-- *Correction: The formula for the sum of roots of \( x^2 - (6+m)x + (10-c) = 0 \) is \( x_1+x_2 = 6+m \). So \( x_{mid} = (6+m)/2 \). Let's re-derive the hack for clarity.* **Corrected Hack:** When you set the equations equal, you get \( x^2 - (6+m)x + (10-c) = 0 \). The sum of the roots \( x_1+x_2 \) is \( -(-(6+m))/1 = 6+m \). The x-coordinate of the midpoint is \( (x_1+x_2)/2 \). Therefore, \( x_{mid} = (6+m)/2 \). Given \( x_{mid} = 4 \): \( 4 = \frac{6 + m}{2} \) Multiply by 2: \( 8 = 6 + m \) Subtract 6: \( m = 2 \) This method allows you to quickly set up and solve for 'm' by directly applying the relationship between the midpoint's x-coordinate and the coefficients of the combined equation.