For what value(s) of \( k \) will the equation \( 3 k x^{2}+5=-5 x \) have non-real roots? INSTRUCTION: Give your answer as an equation or inequality with a fraction (no decimal values). For example: \( \mathrm{k}<7 / 4 \).

To determine the values of \( k \) for which the equation \( 3 k x^{2} + 5 = -5 x \) has non-real roots, we first need to rearrange the equation into standard quadratic form \( ax^2 + bx + c = 0 \). Starting with the given equation: \[ 3 k x^{2} + 5 + 5 x = 0 \] This can be rewritten as: \[ 3 k x^{2} + 5 x + 5 = 0 \] Here, we identify the coefficients: - \( a = 3k \) - \( b = 5 \) - \( c = 5 \) For a quadratic equation to have non-real roots, the discriminant must be less than zero. The discriminant \( D \) is given by: \[ D = b^2 - 4ac \] Substituting the values of \( a \), \( b \), and \( c \): \[ D = 5^2 - 4(3k)(5) \] Calculating this gives: \[ D = 25 - 60k \] To find the values of \( k \) for which the roots are non-real, we set the discriminant less than zero: \[ 25 - 60k < 0 \] Now, we solve this inequality: \[ 25 < 60k \] Dividing both sides by 60: \[ \frac{25}{60} < k \] Simplifying \( \frac{25}{60} \): \[ \frac{5}{12} < k \] Thus, the solution is: \[ k > \frac{5}{12} \] This means the values of \( k \) for which the equation has non-real roots are given by: \[ k > \frac{5}{12} \]