Question

Determine the nature of the roots of the equation $ 4{x^2} - 4x + 1 = 0 $

Answer 1

Hint: The roots of an equation are the points on the x-axis at which the y-coordinate of the function is zero. The roots of an equation can be found out by factoring the equation or by completing the square method. To find the nature of the roots of a quadratic equation, we first find its discriminant. If the discriminant is greater than zero, the roots are real and unequal; if the discriminant is equal to zero then the roots are real and equal and if the discriminant is smaller than zero then the roots are imaginary. Using this information, we can find out the answer to the given question.

Complete step-by-step answer:
We are given that $ 4{x^2} - 4x + 1 = 0 $
Comparing it with the standard form of a quadratic equation that is $ a{x^2} + bx + c = 0 $ , we get –
 $ a = 4,\,b = - 4\,and\,c = 1 $
The discriminant of a quadratic equation can be found out using the formula, $ D = {b^2} - 4ac $
Put the values of a, b and c in the above formula, we get –
 $
  D = {( - 4)^2} - 4 \times 4 \times 1 \\
  D = 16 - 16 \\
   \Rightarrow D = 0 \;
  $
Thus, the roots of the equation $ 4{x^2} - 4x + 1 = 0 $ are real and equal.
So, the correct answer is “The roots of the equation $ 4{x^2} - 4x + 1 = 0 $ are real and equal”.

Note: In a polynomial equation, the highest exponent of the polynomial is called its degree. And according to the Fundamental Theorem of Algebra, a polynomial equation has exactly as many roots as its degree. The roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts. To find the roots or the nature of the roots of an equation, we have to first express the equation in the standard form.