If p(x) = 3x^2 - 18x + 27, how will the graph meet the x-axis?
Answer and explanation
Correct answer: It will touch the x-axis at x = 3
The governing concept is the geometrical meaning of zeroes: a zero is an x-coordinate where the graph meets the x-axis. Factor the polynomial: p(x) = 3x^2 - 18x + 27 = 3(x^2 - 6x + 9) = 3(x - 3)^2. Thus the only zero is x = 3, and it is repeated. A quadratic with a repeated real zero touches the x-axis at that point instead of crossing it. Therefore, option A is correct. Option B has the wrong sign, option C would require two distinct zeroes, and option D would apply only when there are no real zeroes.
Frequently asked questions
What is the correct answer to this question?
It will touch the x-axis at x = 3
Why is this the correct answer?
The governing concept is the geometrical meaning of zeroes: a zero is an x-coordinate where the graph meets the x-axis. Factor the polynomial: p(x) = 3x^2 - 18x + 27 = 3(x^2 - 6x + 9) = 3(x - 3)^2. Thus the only zero is x = 3, and it is repeated. A quadratic with a repeated real zero touches the x-axis at that point instead of crossing it. Therefore, option A is correct. Option B has the wrong sign, option C would require two distinct zeroes, and option D would apply only when there are no real zeroes.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Geometrical meaning of the zeroes of a polynomial..
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