Question

What is the discriminant of ${x^2} + 5x + 5 = 0$.

Answer 1

Hint: For the quadratic equation of the form $a{x^2} + bx + c = 0$ , the discriminant of the given equation is given by $D = {b^2} - 4ac$. The determinant of any quadratic equation is determined to know about the roots of the given quadratic equation.

Complete step by step answer:
We are given the quadratic equation of the form ${x^2} + 5x + 5 = 0$.
Compare it with the standard form of the quadratic equation $a{x^2} + bx + c = 0$, we get the values of a, b and c given by
$a = 1,b = 5,c = 5$
To determine the value of the discriminant we need to substitute the above values of a, b and c in the formulae of the discriminant given by $D = {b^2} - 4ac$.
We get,
$D = {5^2} - 4(1)(4)$
Evaluate the square term,
$D = 25 - 4(1)(4)$
Multiply the term on the right side given by $4 \times 1 \times 4$
$D = 25 - 16$
Now, Solve the above term on the right side to determine the value of the discriminant D.
$D = 9$
So the value of the discriminant is equal to 9. Since the value of the discriminant is a positive integer , the given quadratic equation will have two distinct real roots which are evaluated using the quadratic formulae for solving the quadratic equation. This formula requires the value of the discriminant to evaluate the root.

Note:
If the value of the discriminant is greater than 0, then the roots of the quadratic equation are distinct real numbers. If the discriminant is equal to 0 ,then the quadratic equation has two equal real roots and if the discriminant is less than 0, then the equation has complex roots.
Hence using the quadratic method we solved the given problem.