Question Details

Find the roots of the quadratic equation $15 x^{2} - x - 2 = 0$

Options

$x = \frac{- 1}{3} or x = \frac{- 2}{5}$

$x = \frac{1}{3} or x = \frac{2}{5}$

$x = \frac{- 1}{3} or x = \frac{2}{5}$

$x = \frac{1}{3} or x = \frac{- 2}{5}$

Show Answer

Correct Answer :

Option C

$x = \frac{- 1}{3} or x = \frac{2}{5}$

Solution :

The correct option is:
$x = \frac{- 1}{3} or x = \frac{2}{5}$

To find the roots of the quadratic equation, we start with the given equation:
$15 x^{2} - x - 2 = 0$

We can solve this quadratic equation using the method of splitting the middle term. First, we need to find two numbers whose product is equal to the product of the coefficient of $x^{2}$ and the constant term, and whose sum is equal to the coefficient of the middle term $x$ .
Here, the product is:
$15 \times (- 2) = - 30$
And the sum is:
$- 1$

The two numbers that satisfy these conditions are $- 6$ and $5$ because:
$(- 6) \times 5 = - 30$
and
$- 6 + 5 = - 1$

Now, rewrite the middle term of the equation using these factors:
$15 x^{2} - 6 x + 5 x - 2 = 0$

Group the terms to find common factors:
$(15 x^{2} - 6 x) + (5 x - 2) = 0$

Factor out the common terms from each group:
From the first group $(15 x^{2} - 6 x)$ , we factor out $3 x$ :
$3 x (5 x - 2)$
From the second group $(5 x - 2)$ , we factor out $1$ :
$1 (5 x - 2)$