Question

If alpha and beta are zeroes of the quadratic polynomial f(x)=3x^2-x-4

Answer 1

Question:

If alpha and beta are the zeros of the quadratic polynomial f(x) = 3x² - x - 4. Find the values of alpha and beta.

Answer:

Let $\alpha\:and\: \beta$ be the zeros of the polynomial.

Now, we can factorise the given polynomial f(x) = 3x² - x - 4 by using splitting the middle term method.

3x² - x - 4

Product of co - efficient of x² and the constant term.

$3 \times - 4 = - 12$

Therefore, we've to find those factors of - 12 which gives us the sum - 1.

The number which gives us the product as - 12 and sum as - 1 are - 4 and + 3.

By substituting, - 4 and 3 in the place of - x with variable x along with their signs.

3x² - x - 4

3x² - 4x + 3x - 4

Taking common out.

x(3x - 4) + 1 (3x - 4)

(3x - 4) (x + 1)

Now equating (3x - 4) (x + 1) with 0.

(3x - 4) (x + 1) = 0

(3x - 4) = 0

3x - 4 = 0

3x = 4

$\boxed{\bold{x = \frac{4}{3}}}$

(x + 1) = 0

x + 1 = 0

$\boxed{\bold{x = - 1}}$

Therefore, the value of $\alpha$ and $\beta$ are $\frac{4}{3}$ and - 1.