# Using the discriminant, give the nature of the roots of 7x^3+x^2-35x=5?

## Using the discriminant, give the nature of the roots of $7 {x}^{3} + {x}^{2} - 35 x = 5$. Also solve the equation.

Mar 13, 2017

We have real roots.

#### Explanation:

The discriminant of a cubic equation $a {x}^{3} + b {x}^{2} + c x + d = 0$ is ${b}^{2} {c}^{2} - 4 a {c}^{3} - 4 {b}^{3} d - 27 {a}^{2} {d}^{2} + 18 a b c d$.

• If discriminant is positive, we have real roots;
• if it is zero it has repeated roots and all real; and
• if it is less than zero, we have one rel root and two complex conjugates.

In the given equation $7 {x}^{3} + {x}^{2} - 35 x - 5 = 0$, discriminant is

${1}^{2} \times {\left(- 35\right)}^{2} - 4 \times 7 \times {\left(- 35\right)}^{3} - 4 \times {1}^{3} \times \left(- 5\right) - 27 \times {7}^{2} \times {\left(- 5\right)}^{2} + 18 \times 7 \times 1 \times \left(- 35\right) \times \left(- 5\right)$

= $1225 + 1200500 + 20 - 33075 + 22050 = 1190720$

Hence, we have real roots.