Question

The roots of \[a{x^2} + bx + c = 0,a \ne 0\] are real and unequal, if \[{b^2} - 4ac\] is _________.
A \[ \geqslant 0\]
B \[ > 0\]
C \[ < 0\]
D None of these

Answer 1

Hint:
Quadratic equations can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum one term that is squared. Roots of the quadratic equation can be found by factoring the polynomial and equate it with zero. We know that a second-degree polynomial will have at most two zeros. Therefore, a quadratic equation will have at most two roots.

Complete step by step solution:
The roots of the equation \[a{x^2} + bx + c = 0\] are
\[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
For the equation to have real roots,
\[{b^2} - 4ac \geqslant 0\]
For the equation to have unequal roots,
\[{b^2} - 4ac > 0\]
Hence, option B is correct.
Therefore, the roots of \[a{x^2} + bx + c = 0,a \ne 0\] are real and unequal, if \[{b^2} - 4ac\] is \[ > 0\].

Additional information:
The definite form of quadratic equation is \[a{x^2} + bx + c = 0\]; where x is an unknown variable and a, b, c are numerical coefficients Here, \[a \ne 0\]because if it equals to zero then the equation will not remain quadratic anymore and it will become a linear equation, such as \[bx + c = 0\]. The terms a, b and c are also called quadratic coefficients.
The solutions to the quadratic equation are the values of unknown variables \[x\], which satisfy the equation. These solutions are called roots or zeros of quadratic equations. It means that, if we put the value of roots in the given quadratics, L.H.S. will be equal to R.H.S. of the equation. The roots of any polynomial are the solutions for the given equation.

Note:
Since the quadratic includes only one unknown term or variable, thus it is called univariate. The power of variable \[x\] are always non-negative integers; hence the equation is a polynomial equation with highest power as 2 and there are four different methods to solve the quadratic equation i.e., using factoring, using square roots, completing the squares and using quadratic formula.