Question

How do you evaluate $\sqrt {{b^2} - 4ac} $ for $a = 2$, $b = - 5$, $c = 2$?

Answer 1

Hint: Given the values of the variable a, b and c. We have to find the value of the expression. First, we substitute the values into the expression. Then, we will simplify the expression. Then, check whether the value of radical expression is negative or positive. If positive, then find the square root of the expression, if it is a perfect square. If negative, then the result of the square root is in the form of a complex number.

Complete step by step solution:
We are given the expression, $\sqrt {{b^2} - 4ac} $.

Substitute $a = 2$, $b = - 5$, $c = 2$ into the formula, we get:

$ \Rightarrow \sqrt {{{\left( { - 5} \right)}^2} - 4 \times 2 \times 2} $

On simplifying the terms of the expression, we get:

$ \Rightarrow \sqrt {25 - 16} $

On further simplifying the expression, we get:

$ \Rightarrow \sqrt 9 $

Determine the square root of the number.

$ \Rightarrow \sqrt {{b^2} - 4ac} = \pm 3$

Final answer: Hence the value of the expression, $\sqrt {{b^2} - 4ac} $ is $ \pm 3$

Note: Please note that the nature of the roots of the equation will depend on the value of discriminant, where the expression ${b^2} - 4ac$ is known as discriminant of the quadratic equation. If the value of discriminant is less than zero, then the equation has no real roots which means it has only complex roots. If the value of discriminant is equal to zero, then the equation has two equal real roots. On the other hand if the value of discriminant is greater than zero then the equation has two real roots which are distinct. Here, on squaring the value of $\sqrt {{b^2} - 4ac} $, we get the discriminant ${b^2} - 4ac$ which is equal to 9. So, the discriminant is greater than zero. This means the quadratic equation can have two distinct real roots.