Question
Use the discriminant to determine the number of x-intercepts for the following equations.
a) 𝑦 = 5𝑥^2 + 6𝑥 + 6
b) 10 − 4𝑥 − 3𝑥^2 = 0
a) 𝑦 = 5𝑥^2 + 6𝑥 + 6
b) 10 − 4𝑥 − 3𝑥^2 = 0
Answers
Answered by
GPT 3.5
a) The discriminant can be calculated using the formula:
𝐷 = 𝑏^2 - 4𝑎𝑐
For the equation 𝑦 = 5𝑥^2 + 6𝑥 + 6,
𝑎 = 5, 𝑏 = 6, 𝑐 = 6
𝐷 = (6)^2 - 4(5)(6)
𝐷 = 36 - 120
𝐷 = -84
Since the discriminant is negative (-84), this means that the equation 𝑦 = 5𝑥^2 + 6𝑥 + 6 has no x-intercepts.
b) The equation 10 − 4𝑥 − 3𝑥^2 = 0 can be rearranged to the form:
𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0
For this equation, 𝑎 = -3, 𝑏 = -4, and 𝑐 = 10.
𝐷 = 𝑏^2 - 4𝑎𝑐
𝐷 = (-4)^2 - 4(-3)(10)
𝐷 = 16 + 120
𝐷 = 136
Since the discriminant is positive (136), this means that the equation 10 − 4𝑥 − 3𝑥^2 = 0 has two x-intercepts.
𝐷 = 𝑏^2 - 4𝑎𝑐
For the equation 𝑦 = 5𝑥^2 + 6𝑥 + 6,
𝑎 = 5, 𝑏 = 6, 𝑐 = 6
𝐷 = (6)^2 - 4(5)(6)
𝐷 = 36 - 120
𝐷 = -84
Since the discriminant is negative (-84), this means that the equation 𝑦 = 5𝑥^2 + 6𝑥 + 6 has no x-intercepts.
b) The equation 10 − 4𝑥 − 3𝑥^2 = 0 can be rearranged to the form:
𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0
For this equation, 𝑎 = -3, 𝑏 = -4, and 𝑐 = 10.
𝐷 = 𝑏^2 - 4𝑎𝑐
𝐷 = (-4)^2 - 4(-3)(10)
𝐷 = 16 + 120
𝐷 = 136
Since the discriminant is positive (136), this means that the equation 10 − 4𝑥 − 3𝑥^2 = 0 has two x-intercepts.