If p(x)=x^2+2x−8 and q(x)=x^2+2x−7, which comparison is correct?
Answer and explanation
Correct answer: p(x) has rational zeroes and q(x) has irrational real zeroes
For ax²+bx+c, the discriminant D=b²−4ac determines the nature of the zeroes. For p(x), D=2²−4(1)(−8)=4+32=36, a positive perfect square. Hence p has two distinct rational real zeroes, namely 2 and −4. For q(x), D=2²−4(1)(−7)=4+28=32, which is positive but not a perfect square. Hence q has two distinct irrational real zeroes. Both quadratics therefore have real zeroes, but only p’s are rational. This makes option A correct. Options B and D reverse the classifications, while C incorrectly treats positive discriminants as non-real.
Frequently asked questions
What is the correct answer to this question?
p(x) has rational zeroes and q(x) has irrational real zeroes
Why is this the correct answer?
For ax²+bx+c, the discriminant D=b²−4ac determines the nature of the zeroes. For p(x), D=2²−4(1)(−8)=4+32=36, a positive perfect square. Hence p has two distinct rational real zeroes, namely 2 and −4. For q(x), D=2²−4(1)(−7)=4+28=32, which is positive but not a perfect square. Hence q has two distinct irrational real zeroes. Both quadratics therefore have real zeroes, but only p’s are rational. This makes option A correct. Options B and D reverse the classifications, while C incorrectly treats positive discriminants as non-real.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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