If \(p(x)=x^2+6x+10\), how does its graph relate to the x-axis?
Answer and explanation
Correct answer: It will not meet the \(x\)-axis
Completing the square gives \(x^2+6x+10=(x+3)^2+1\). Since \((x+3)^2\ge0\), the minimum value is 1, so the parabola never reaches 0 and does not meet the x-axis. Equivalently, the discriminant is \(b^2-4ac=36-40=-4<0\), so there are no real roots. The closest distractor (touch once) is wrong because that requires discriminant zero. Exam tip: use the discriminant or complete the square to quickly decide if real x-intercepts exist.
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What is the correct answer to this question?
It will not meet the \(x\)-axis
Why is this the correct answer?
Completing the square gives \(x^2+6x+10=(x+3)^2+1\). Since \((x+3)^2\ge0\), the minimum value is 1, so the parabola never reaches 0 and does not meet the x-axis. Equivalently, the discriminant is \(b^2-4ac=36-40=-4<0\), so there are no real roots. The closest distractor (touch once) is wrong because that requires discriminant zero. Exam tip: use the discriminant or complete the square to quickly decide if real x-intercepts exist.
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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