Question

Find the sum and product of the roots of the quadratic equation ${x^2} - 5x + 8 = 0$.

Answer 1

Hint: The given equation is a quadratic equation. The sum and product of the roots of the quadratic equation can be calculated by using a formula that is:
Sum of the roots $ = \dfrac{{ - b}}{a}$
Product of the roots $ = \dfrac{c}{a}$
Where $a$ is the coefficient of ${x^2}$, $b$ is the coefficient of $x$ and $c$ is the constant term of a quadratic equation $a{x^2} + bx + c = 0$.

Complete step-by-step solution:
The given quadratic equation is ${x^2} - 5x + 8 = 0$.
Now, comparing the given quadratic equation with the general form of a quadratic equation. We get,
$a = 1$ , $b = - 5$ and $c = 8$.
Now, applying the formula for the sum of the roots. we get,
Sum of the roots $ = \dfrac{{ - b}}{a} = \dfrac{{ - \left( { - 5} \right)}}{1} = 5$
And, applying the formula for the product of the roots. we get.
Product of the roots $ = \dfrac{c}{a} = \dfrac{8}{1} = 8$.

Thus, the sum of roots of the given quadratic equation is $5$ and the product of the roots is $8$.

Note: The discriminant of a quadratic equation $D = {b^2} - 4ac$. If the discriminant of a quadratic equation is negative then the roots will be imaginary. If discriminant is zero then both the roots are equal and if the discriminant is positive then both the roots are real and distinct.
Now, the discriminant of the given quadratic equation $D = {\left( 5 \right)^2} - 4 \times 1 \times 8 = 25 - 32 = - 7$.
Since the discriminant is negative so, the roots of the given quadratic equation are imaginary.
If the given equation is cubic that is $a{x^3} + b{x^2} + cx + d = 0$. Then,
The sum of the roots of the cubic equation is given by $\dfrac{{ - b}}{a}$.
The product of the roots is given by $\dfrac{{ - d}}{a}$.