Question

Assertion: If $a > 0$ , the value of the expression ${b^2} - 4ac$ will be greater than zero.
Reason: The expression ${b^2} - 4ac$ represents the discriminant of a quadratic function. If $a > 0$ the graph will be an upward parabola and it will cut the $x$- axis at two distinct points indicating real and distinct roots. For real and distinct roots, the value of the discriminant is greater than zero.
A. Both Assertion and Reason are correct and Reason is the correct explanation for assertion.
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Both Assertion and Reason are incorrect.

Answer 1

Hint: The question can be understood and solved with the help of a clear understanding of the Quadratic Equation and its Formula. The graphical representation of a quadratic equation is a parabola which depends on the value of $a$ and its sign and also decides whether the parabola will be an upward-facing parabola or a downward-facing parabola.

Complete step by step solution:
The assertion given in the question is
 If $a > 0$ , the value of the expression ${b^2} - 4ac$ will be greater than zero.
Now, the quadratic equation is written as $a{x^2} + bx + c = 0$ where $a$ , $b$ and c represent known numbers and $a \ne 0$ .
The solution of the above equation can be given by the formula $x = \dfrac{{ - b \pm \sqrt {b{}^2 - 4ac} }}{{2a}}$ . The expression under the square root sign is called discriminant of the quadratic equation. A quadratic equation with real coefficients of the variables can have either one or two distinct real roots or two distinct complex roots. The discriminant also determines the number and nature of roots.
There are two distinct roots if the discriminant is positive. If the discriminant is zero, then there is exactly one real root and if the discriminant is negative, then there are no real roots.
If $a > 0$ , then the parabola has a minimum point and opens upward. If $a < 0$ then the parabola has a maximum point and opens downward.
The above explanation proves that the reason given in the question is correct, that is:
The expression ${b^2} - 4ac$ represents the discriminant of a quadratic function. If $a > 0$ the graph will be an upward parabola and it will cut the $x$- axis at two distinct points indicating real and distinct roots. For real and distinct roots, the value of the discriminant is greater than zero.

Therefore, option A, that is,” Both Assertion and Reason are correct and Reason is the correct explanation for assertion” is the correct option.

Note: The solutions of the quadratic equation $a{x^2} + bx + c = 0$ gives the roots of the function $f(x) = a{x^2} + bx + c$ since they are the values of the variable $x$ for which the function $f(x) = 0$ . While representing the equation on the graph if the discriminant is positive, the graph will touch the $x$- axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the $x$- axis.