Factorising the quadratic equation 3x^2 - 14x - 5 = 0

To factorise the given quadratic equation 3x^2 - 14x - 5 = 0, we need to find two binomials whose product is equal to the quadratic equation. The process of factorisation involves breaking down the quadratic expression into its factors.

Step 1: Multiply the coefficient of x^2 and the constant term
In this case, the coefficient of x^2 is 3, and the constant term is -5. When we multiply these two values, we get -15.

Step 2: Find two numbers whose product is equal to -15 and whose sum is equal to the coefficient of x
To find these numbers, we need to consider all the possible pairs of factors of -15. The pairs of factors of -15 are:
-15 and 1
15 and -1
-3 and 5
3 and -5

Among these pairs, the pair whose sum is equal to the coefficient of x (-14) is -3 and 5. Therefore, we can rewrite the middle term (-14x) as -3x + 5x.

Step 3: Group the terms
Now, we can rewrite the quadratic equation as follows:
3x^2 - 3x + 5x - 5 = 0

Step 4: Factor by grouping
Next, we group the terms into two pairs:
(3x^2 - 3x) + (5x - 5) = 0

Step 5: Factor out the common factors
From the first pair, we can factor out a common term of x:
3x(x - 1)

From the second pair, we can factor out a common term of 5:
5(x - 1)

Step 6: Write the factored form
Combining the factored terms, we get:
(3x + 5)(x - 1) = 0

Therefore, the factored form of the quadratic equation 3x^2 - 14x - 5 = 0 is (3x + 5)(x - 1) = 0.

Conclusion:
The quadratic equation 3x^2 - 14x - 5 = 0 can be factorised as (3x + 5)(x - 1) = 0. Factoring the quadratic equation helps us find its roots or solutions, which are the values of x that satisfy the equation. In this case, the solutions are x = -5/3 and x = 1.

-3 and 5 is -14 how ?