If (x^2-10x+k) is a perfect square, what is the value of (k)?
Answer and explanation
Correct answer: (25)
A quadratic of the form \(x^2+bx+k\) is a perfect square when it matches \((x-r)^2=x^2-2rx+r^2\). Comparing \(x^2-10x+k\) with this identity gives \(-2r=-10\), so \(r=5\). The constant term must then be \(r^2=25\). Thus the expression is \((x-5)^2\), and option C is correct.
The same result follows from completing the square: half of the coefficient of \(x\) is \(-10/2=-5\), and its square is \((-5)^2=25\). Therefore \(x^2-10x+25=(x-5)^2\). The sign disappears after squaring, but the middle term must remain negative. Options A, B, and D do not produce the required middle coefficient and constant term together.
Frequently asked questions
What is the correct answer to this question?
(25)
Why is this the correct answer?
A quadratic of the form \(x^2+bx+k\) is a perfect square when it matches \((x-r)^2=x^2-2rx+r^2\). Comparing \(x^2-10x+k\) with this identity gives \(-2r=-10\), so \(r=5\). The constant term must then be \(r^2=25\). Thus the expression is \((x-5)^2\), and option C is correct.
The same result follows from completing the square: half of the coefficient of \(x\) is \(-10/2=-5\), and its square is \((-5)^2=25\). Therefore \(x^2-10x+25=(x-5)^2\). The sign disappears after squaring, but the middle term must remain negative. Options A, B, and D do not produce the required middle coefficient and constant term together.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Quadratic expressions.
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