Question

What term is \[{{b}^{2}}-4ac\]?

Answer 1

Hint: For solving this question you should know about the quadratic equations and to find the roots of them. This is \[{{b}^{2}}-4ac\] is not a term it is a part of the quadratic formula which is denoted at place of discriminant. Find this discriminant here denoted as ‘D’. If we find the roots of any quadratic equation, then we use this formula.

Complete step-by-step solution:
According to our question you have to explain what \[{{b}^{2}}-4ac\] term is.
So, as we know that the quadratic equations of the format \[a{{x}^{2}}+bx+c\] always contain roots which can be imaginary or real roots. But in many equations we can’t determine roots easily. So, then we use the formula for finding the roots. If the equation is in \[a{{x}^{2}}+bx+c\] form then Quadratic formula: - \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
But Discriminant ‘D’ = \[{{b}^{2}}-4ac\]
So, we can write it as:
Quadratic formula: \[\dfrac{-b\pm \sqrt{D}}{2a}\]
And here we get two roots as:
\[\dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}\] and \[\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}\]
If we take an example to understand it clearly then: eg(1) Find the roots of \[{{x}^{2}}-4x+6\].
Soln - The discriminant used to determine how many different solutions and what type of solutions a quadratic equation will have.
So, here according to our equation:
\[\begin{align}
  & a=1,b=-4,c=6 \\
 & \Rightarrow D={{\left( -4 \right)}^{2}}-4\left( 1 \right)\left( 6 \right) \\
 & \Rightarrow D=16-24 \\
 & \Rightarrow D=-8 \\
\end{align}\]
This indicates that this equation contains 2 imaginary solutions (with i, the square root of -1)
If the answer had been positive (assume, 9), then the equation would have 2 real solutions (with real numbers; they might not be rational solutions, but they are real).
And if the answer was 0, then the equation has 1 real solution (it would be the square root of something).
Thus, \[{{b}^{2}}-4ac\] is not a term.

Note: If we want to calculate the roots of any equation then always try to reduce that in a form of \[a{{x}^{2}}+bx+c\] because it will be very easy to find the roots from this. And it gives us exact roots. So, always follow this method.