Maths-
General
Easy

Question

y=x2 - a
In the equation above, a is a positive constant and the graph of the equation in the xy‑plane is a parabola. Which of the following is an equivalent form of the equation?

  1. y equals left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis
  2. y equals left parenthesis x plus square root of a right parenthesis left parenthesis x minus square root of a right parenthesis
  3. y equals open parentheses x plus a over 2 close parentheses open parentheses x minus a over 2 close parentheses
  4. y equals left parenthesis x plus a right parenthesis squared

The correct answer is: y equals left parenthesis x plus square root of a right parenthesis left parenthesis x minus square root of a right parenthesis


    We have given the equation y = x2 – a
    The given equation can be thought of as the difference of two squares,
    where one square is x2 and the other square is a.
    We know that , a2-b2= (a + b) (a-b) ,
    Using this difference of squares formula,
    y equals open parentheses x squared minus left parenthesis square root of a right parenthesis squared close parentheses
    Simplifying it we get,
    y equals left parenthesis x plus square root of a right parenthesis left parenthesis x minus square root of a right parenthesis.
    Options A, C, and D are incorrect because they are not equivalent to the given equation.
    Option A is incorrect because it is equivalent to y = x2 a2 .
    Option C is incorrect because it is equivalent to y = x2 – (a2/4).
    Option D is incorrect because it is equivalent to y = x2 + 2ax + a2.

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