Q. 3

In Problems $1 - 6$ , solve each equation.

$2 x^{3} - 3 x^{2} - 8 x - 3 = 0$ .

Verified

Answer

The solutions for the equation $2 x^{3} - 3 x^{2} - 8 x - 3 = 0$ are, $x = - 1, - \frac{1}{2}, 3$ .

1Step 1 Given equation is,

$2 x^{3} - 3 x^{2} - 8 x - 3 = 0$ .

The given equation can be solved by assuming a value for $x$ which is a solution of the given cubic equation in $x$ .

Substitute $x = - 1$ in the given equation.

$2 {(- 1)}^{3} - 3 {(- 1)}^{2} - 8 (- 1) - 3 = 0 - 2 - 3 + 8 - 3 = 0 - 5 + 5 = 0 0 = 0$

Since $0 = 0$ is a true statement, the value $x = - 1$ is a solution to the given equation.

2Step 2 The given cubic equation is a product of x + 1 and a quadratic equation in x .

$2 x^{3} - 3 x^{2} - 8 x - 3 = (x + 1) (a x^{2} + b x + c)$ .

The quadratic equation can be obtained by dividing the cubic equation by $(x + 1)$ .

3Step 3 The obtained equation can now be solved by using the formula.

$x = \frac{5 \pm \sqrt{{(- 5)}^{2} - 4 (2) (- 3)}}{2 (2)} = \frac{5 \pm 7}{4} = - \frac{1}{2}, 3$

Hence the required solution set is, $\{- 1, - \frac{1}{2}, 3\}$ .