Give a value of \( c \) in the equation \( 9 x^{2}-12 x+c=0 \) so we have a. no real roots b. one real root c. two real rational roots (give fractions as a solution) d. two real integer roots (give whole numbers as a solution)

To determine the value of \( c \) in the quadratic equation \( 9x^2 - 12x + c = 0 \) for different types of roots, we can use the discriminant \( D \) of the quadratic formula, which is given by: \[ D = b^2 - 4ac \] In our case, \( a = 9 \), \( b = -12 \), and \( c \) is the unknown. The discriminant will help us find the conditions for the types of roots: 1. **No real roots**: This occurs when \( D < 0 \). 2. **One real root**: This occurs when \( D = 0 \). 3. **Two real rational roots**: This occurs when \( D > 0 \) and \( D \) is a perfect square. 4. **Two real integer roots**: This occurs when \( D > 0 \) and the roots are integers. Let's analyze each case step by step. ### a. No real roots For no real roots, we need: \[ D < 0 \] Calculating the discriminant: \[ D = (-12)^2 - 4 \cdot 9 \cdot c = 144 - 36c \] Setting the inequality: \[ 144 - 36c < 0 \] Solving for \( c \): \[ 144 < 36c \implies c > 4 \] ### b. One real root For one real root, we need: \[ D = 0 \] Setting the discriminant to zero: \[ 144 - 36c = 0 \] Solving for \( c \): \[ 144 = 36c \implies c = 4 \] ### c. Two real rational roots For two real rational roots, we need: \[ D > 0 \quad \text{and} \quad D \text{ is a perfect square} \] Setting the discriminant greater than zero: \[ 144 - 36c > 0 \] Solving for \( c \): \[ 144 > 36c \implies c < 4 \] Now, we need \( D \) to be a perfect square. Let's express \( D \): \[ D = 144 - 36c \] We can set \( D = k^2 \) for some integer \( k \): \[ 144 - 36c = k^2 \implies 36c = 144 - k^2 \implies c = 4 - \frac{k^2}{36} \] To ensure \( c < 4 \), \( k^2 \) must be positive. Let's choose \( k = 6 \): \[ c = 4 - \frac{6^2}{36} = 4 - 1 = 3 \] ### d. Two real integer roots For two real integer roots, we need \( D > 0 \) and the roots must be integers. The roots of the quadratic can be expressed as: \[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{D}}{18} \] For the roots to be integers, \( \sqrt{D} \) must be a multiple of 18. Let’s set \( D = k^2 \) where \( k \) is a multiple of 18. Let \( k = 18 \): \[ D = 18^2 = 324 \] Setting \( D \): \[ 144 - 36c = 324 \implies 36c = 144 - 324 \implies 36c = -180 \implies c = -5 \] ### Summary of Values for \( c \): - a. No real roots: \( c > 4 \) (e.g., \( c = 5 \)) - b. One real root: \( c = 4 \) - c. Two real rational roots: \( c = 3 \) - d. Two real integer roots: \( c = -5 \) These values satisfy the conditions for the types of roots specified.