Question

What is the discriminant of ${x^2} - 5x = 6$ and what does that mean?

Answer 1

Hint: To solve this question we first try to compare it to a standard quadratic equation. After finding the value of a,b and c we will put it into the discriminant formula and calculate the value of discriminant.

Complete step by step answer:
To find the discriminant of quadratic equation we use the formulae of discriminate which is
$D = {b^2} - 4ac$
The given equation is ${x^2} - 5x = 6$
We can write it as ${x^2} - 5x - 6$
Comparing it with the standard equation: $a{x^2} + bx + c$
Here $b = - 5$ and $a = 1,c = - 6$
Keeping value in it. we get,
$D = {( - 5)^2} - 4 \times 1 \times - 6$
By solving it. We get,
$D = 49$
The discriminant of the given quadratic equation is $49$
We know that If D is more than 0, the equation has two real solutions. If D is less than 0, there are no real solutions. If D is equal to 0, there is one real solution.
Here $D > 0$ which means that the equation has two real roots.

Note:
In the case of a quadratic equation $a{x^2} + bx + c = 0$ , the discriminant is ${b^2} - 4ac$ and for a cubic equation ${x^3} + a{x^2} + bx + c = 0$ the discriminant is ${a^2}{b^2} + 18abc - 4{b^3} - 4{a^3}c - 24{c^2}$ .
Sometimes students get confused between the discriminant formula whether it is ${b^2} - 4ac$ or it is $\sqrt{{b^2} - 4ac}$.