Logo for LIVE by Po-Shen Loh

LIVE

English

1957 AMC 12 Problem 40

Attempt Problem 40 of the 1957 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1957 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

40.

If the parabola y=−x2+bx−8y=-x^2+bx-8 has its vertex on the xx-axis, then bb must be:

a positive integer

a positive or a negative rational number

a positive rational number

a positive or a negative irrational number

a negative irrational number

Answer: D
Concepts:quadraticparabolaradical
Difficulty rating: 1360
Small Hint:

A parabola whose vertex lies on the xx-axis has a repeated root

Big Hint:

Set the discriminant of −x2+bx−8=0-x^2+bx-8=0 equal to zero

Solution:

The vertex lies on the xx-axis exactly when the quadratic has one repeated root. Its discriminant must satisfy b2−4(−1)(−8)=b2−32=0. b^2-4(-1)(-8)=b^2-32=0. Hence b=±32=±42,b=\pm\sqrt{32}=\pm4\sqrt2, a positive or negative irrational number.

Thus, the correct answer is D.

← Problem 39#39
Full Exam
Problem 41#41 →

Problem 40 in Other Years

1950 AMC 12 · 1951 AMC 12 · 1952 AMC 12 · 1953 AMC 12 · 1954 AMC 12 · 1955 AMC 12 · 1956 AMC 12 · 1958 AMC 12 · 1959 AMC 12 · 1960 AMC 12 · 1961 AMC 12 · 1962 AMC 12 · 1963 AMC 12 · 1964 AMC 12 · 1965 AMC 12 · 1966 AMC 12 · 1967 AMC 12