Let's try using the discriminant

Algebra Level 1

f ( x ) = a x 2 + b x + c f(x)=ax^2+bx+c

If b > 10 , 0 < a < 2 , c = 3 b>10, 0<a<2, c=3 , how many times does the graph y = f ( x ) y=f(x) cross the x-axis?

Cannot be determined 2 1 0

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Margaret Zheng by Margaret Zheng , 20, USA

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1 solution

Margaret Zheng
Jan 4, 2016

Substitute 0 for f(x), and the discriminant of a quadratic equation is b^2-4ac. Since b>10, b^2>100. Since 0<a<2, ac<6 and 4ac<24. A number greater than 100 subtracted by a number less than 24 is always positive; therefore, the discriminant has a positive value and the graph crosses the x-axis twice.

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But f(x) could be a whole square and cross the x axis only once, hence the correct answer should be 2 or 1

Sarvesh Lanke - 3 years, 9 months ago

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I believe that for what is given in this question, which is b > 10, 0 < a < 2, and c = 3, it is impossible to make f(x) a "whole square"... to do what I think you have described with b > 10, the least c could be is something greater than 25. (i.e. y = x 2 + 10 x + 25 y = x^{2} + 10x + 25 would be a perfect square.) The point of this question is to find the number of solutions using discriminant, which is b 2 − 4 a c b^{2} - 4ac for any quadratic equation. I may be wrong. Could you think of an equation that satisfies all given conditions and still end up being a perfect square?

Margaret Zheng - 3 years, 9 months ago

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