If (p(x)=x^2-4kx+4k^2), at which (x)-value will the graph touch the (x)-axis?
Answer and explanation
Correct answer: (x=2k)
The graph touches the x-axis when the polynomial has a repeated zero. Here, the expression can be rewritten as a perfect square: \(p(x)=x^2-4kx+4k^2=(x-2k)^2\). A square is zero only when its inside expression is zero, so \(x-2k=0\), giving \(x=2k\). Thus the graph has one repeated intercept and touches the x-axis at the point whose x-coordinate is \(2k\), not at \(k\), \(-2k\), or \(4k\). Recognising the perfect-square form is the quickest method.
Equivalently, the quadratic formula gives a discriminant of \((-4k)^2-4(1)(4k^2)=0\), confirming that both zeroes coincide. Their common value is \(\frac{4k}{2}=2k\). Therefore option B is correct. The graph does not cross the axis at this repeated zero; it merely touches it, because \((x-2k)^2\) is never negative.
Frequently asked questions
What is the correct answer to this question?
(x=2k)
Why is this the correct answer?
The graph touches the x-axis when the polynomial has a repeated zero. Here, the expression can be rewritten as a perfect square: \(p(x)=x^2-4kx+4k^2=(x-2k)^2\). A square is zero only when its inside expression is zero, so \(x-2k=0\), giving \(x=2k\). Thus the graph has one repeated intercept and touches the x-axis at the point whose x-coordinate is \(2k\), not at \(k\), \(-2k\), or \(4k\). Recognising the perfect-square form is the quickest method.
Equivalently, the quadratic formula gives a discriminant of \((-4k)^2-4(1)(4k^2)=0\), confirming that both zeroes coincide. Their common value is \(\frac{4k}{2}=2k\). Therefore option B is correct. The graph does not cross the axis at this repeated zero; it merely touches it, because \((x-2k)^2\) is never negative.
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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