Let's break down each question and solve them step by step: --- ### 1. **Find the value of the discriminant to determine the number of zeros** For a quadratic equation of the form \( ax^2 + bx + c \), the discriminant \( D \) is: \[ D = b^2 - 4ac \] - **If \( D > 0 \)**: 2 real roots - **If \( D = 0 \)**: 1 real root - **If \( D < 0 \)**: No real roots #### a. \( f(x) = 5x^2 - 4x + 2 \) - \( a = 5, b = -4, c = 2 \) \[ D = (-4)^2 - 4(5)(2) = 16 - 40 = -24 \] **Number of zeros:** 0 real roots (no real roots) #### b. \( f(x) = 9x^2 - 14.4x + 5.76 \) - \( a = 9, b = -14.4, c = 5.76 \) \[ D = (-14.4)^2 - 4(9)(5.76) = 207.36 - 207.36 = 0 \] **Number of zeros:** 1 real root --- ### 2. **Find the number of roots for the equation \(-7x^2 - 56x - 112 = 0\)** - \( a = -7, b = -56, c = -112 \) \[ D = (-56)^2 - 4(-7)(-112) = 3136 - 3136 = 0 \] **Number of roots:** 1 real root --- ### 3. **Without graphing, say whether the graph of each function intersects the x-axis at one point, two points, or does not intersect the x-axis at all.** #### a. \( f(x) = 16 + 8x + x^2 \) Standard form: \( x^2 + 8x + 16 \) - \( a = 1, b = 8, c = 16 \) \[ D = 8^2 - 4(1)(16) = 64 - 64 = 0 \] **Intersects x-axis:** Once (1 root) #### b. \( f(x) = 2x^2 + 4x - 3 \) - \( a = 2, b = 4, c = -3 \) \[ D = 4^2 - 4(2)(-3) = 16 + 24 = 40 \] **Intersects x-axis:** Twice (2 roots) --- ### 4. **The function \( g(x) = 4x^2 - 4x + m \) has exactly one zero. What is the value of \( m \)?** - \( a = 4, b = -4, c = m \) - For exactly one zero, \( D = 0 \): \[ (-4)^2 - 4(4)m = 0 \\ 16 - 16m = 0 \\ 16m = 16 \implies m = 1 \] --- ### 5. **For what values of \( k \) does the equation \( 3x^2 + 4x + k = 0 \) have (a) no real roots, (b) one real root, (c) two real roots?** - \( a = 3, b = 4, c = k \) - Discriminant: \( D = 4^2 - 4(3)k = 16 - 12k \) **a. No real roots:** \( D < 0 \) \[ 16 - 12k < 0 \implies 12k > 16 \implies k > \frac{4}{3} \] **b. One real root:** \( D = 0 \) \[ 16 - 12k = 0 \implies 12k = 16 \implies k = \frac{4}{3} \] **c. Two real roots:** \( D > 0 \) \[ 16 - 12k > 0 \implies 12k < 16 \implies k < \frac{4}{3} \] --- Let me know if you need explanations for any of the steps!