Factor the trinomial [tex]7x^2 - 3x - 4[/tex].

1. Which pair of numbers has a product of [tex]ac[/tex] and a sum of [tex]b[/tex]?
[tex]\[\square\][/tex]

2. What is the factored form of the trinomial?
[tex]\[\square\][/tex]



Answer :

To factor the trinomial [tex]\(7x^2 - 3x - 4\)[/tex], follow these steps:

1. Identify coefficients:
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -4\)[/tex]

2. Calculate the product [tex]\(ac\)[/tex]:
- [tex]\( ac = 7 \times (-4) = -28 \)[/tex]

3. Find two numbers that multiply to [tex]\(ac\)[/tex] and add to [tex]\(b\)[/tex]:
- We need two numbers that multiply to [tex]\(-28\)[/tex] and add to [tex]\(-3\)[/tex].
- These numbers are [tex]\(4\)[/tex] (since [tex]\(4 + (-7) = -3\)[/tex] and [tex]\(4 \times (-7) = -28\)[/tex]).

4. Split the middle term using these numbers:
- Rewrite the trinomial: [tex]\(7x^2 - 3x - 4\)[/tex] becomes [tex]\(7x^2 + 4x - 7x - 4\)[/tex].

5. Factor by grouping:
- Group the terms: [tex]\((7x^2 + 4x) + (-7x - 4)\)[/tex].
- Factor out the common factors in each group:
- For the first group: [tex]\(7x(x + \frac{4}{7}) \)[/tex]
- For the second group: [tex]\(-1(7x + 4)\)[/tex].

6. Combine the common factors:
- Notice that both groups have a common factor of [tex]\((x + \frac{4}{7})\)[/tex].
- The factored expression is: [tex]\( (x + \frac{4}{7})(7x - 1) \)[/tex].

However, to see the typical form for integer pairs, it simplifies directly to:

[tex]\((x - 1)(7x + 4)\)[/tex].

So, the factored form of the trinomial [tex]\(7x^2 - 3x - 4\)[/tex] is:

[tex]\[ (x - 1)(7x + 4) \][/tex]