Question

What do you mean by the range of a Quadratic Equation.
А) The range of values that the equation can have
B) Values of the term with coefficient
C) Values of the Constant term
D) None of the above

Answer 1

Hint: Range of a function is the set of all real values in the given domain. Similarly, the range of a quadratic equation is the value of all the real numbers we get by solving the equation. So, by checking the given multiple choices, we will check the given choices, hence we will get our required answer.

Complete step by step solution:
We have been given four multiple choices here, we have to choose which one is true regarding the given question, i.e., range of a Quadratic Equation.
So, we know that quadratic equation is an equation is in a form of \[a{x^2} + bx + c{\text{ }} = {\text{ }}0,\] where a, b and c are the values which are known to us. Here, ‘a’ can't be equal to \[0,\] ‘x’ is the variable.
These equations are called quadratic equations because ‘quad’ means square, since here the variable gets squared, like \[{x^2}\]
The range of quadratic equations can be defined as the values that we get after solving the equation in the provided domain.
Now, let us check the given options one by one, by considering the standard form of quadratic equation, i.e., \[a{x^2} + {\text{ }}bx{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0\]
The first option is, the range of values that the equation can have, true, the range gives the values of the equation, so this option is correct.
The second option is, values of the term with coefficient, false, values of the equation is the range of quadratic equation, but not with the coefficient, so this option is incorrect.
The third option is, values of the Constant term, false, values of the equation is the range of quadratic equation, but not of the constant term but of the variable, so this option is incorrect.
The fourth option is, none of the above, false, since we already get our answer, so this option is incorrect.
Thus, option (A) is correct.
So, the correct answer is “Option A”.

Note: Let us take an example of finding the range of a quadratic equation.
So, let the quadratic equation be, \[y = 2{x^2} - 3x + 1\] when
Considering, \[2{x^2} - 3x + 1\; = 0\]
Now,
  \[
  2{x^2} - 2x - x + 1 = 0 \\
   \Rightarrow 2x\left( {x - 1} \right)-1\left( {x - 1} \right) = 0 \\
   \Rightarrow (x - 1)(2x - 1) = 0 \\
   \Rightarrow x = 1,\dfrac{1}{2} \;
 \]
So, the range of y is, $y\left[ {1,\dfrac{1}{2}} \right] $