If the quadratic function 3x² - 7x + R is a perfect square, find R.
A. 49/24 B. 49/12 C. 49/13 D. 49/3
Correct Answer: A
Explanation
To determine the value of \( R \) for which the quadratic function \( 3x^2 - 7x + R \) is a perfect square, follow these steps:
1. Rewrite the Quadratic Function: A quadratic function \( ax^2 + bx + c \) is a perfect square if it can be expressed as \( (mx + n)^2 \). For \( 3x^2 - 7x + R \) to be a perfect square, it must be of the form: \[ (mx + n)^2 \]
2. Expand the Perfect Square Expression: Expand \( (mx + n)^2 \) and equate it to \( 3x^2 - 7x + R \): \[ (mx + n)^2 = m^2x^2 + 2mnx + n^2 \] Compare this with \( 3x^2 - 7x + R \): \[ m^2 = 3 \\ 2mn = -7 \\ n^2 = R \]
3. Solve for \( m \) and \( n \): - To satisfy \( m^2 = 3 \): \[ m = \pm \sqrt{3} \] - Using \( 2mn = -7 \): \[ 2(\sqrt{3})n = -7 \\ n = \frac{-7}{2\sqrt{3}} = \frac{-7\sqrt{3}}{6} \] or \[ 2(-\sqrt{3})n = -7 \\ n = \frac{7}{2\sqrt{3}} = \frac{7\sqrt{3}}{6} \] - Calculate \( R \): \[ n^2 = \left(\frac{-7\sqrt{3}}{6}\right)^2 = \frac{49 \times 3}{36} = \frac{147}{36} = \frac{49}{12} \] or \[ n^2 = \left(\frac{7\sqrt{3}}{6}\right)^2 = \frac{49 \times 3}{36} = \frac{147}{36} = \frac{49}{12} \]
The value of \( R \) that makes \( 3x^2 - 7x + R \) a perfect square is \( \frac{49}{12} \).