Factoring Quadratics when Coefficient > 1 (examples, solutions ... - Free Printable

Explanation:
We are given the quadratic expression:
$$ 6x^2 + 5x - 4 $$
and we want to factor it without guessing, using the method shown.

The method works like this:

1. Multiply the leading coefficient $ a = 6 $ and the constant term $ c = -4 $:
$$
ac = 6 \times (-4) = -24
$$

2. Find two numbers that:
- Multiply to $ -24 $
- Add to the middle coefficient $ b = 5 $

Let’s list factor pairs of $-24$:

- $1$ and $-24$: sum = $-23$ ✘
- $-1$ and $24$: sum = $23$ ✘
- $2$ and $-12$: sum = $-10$ ✘
- $-2$ and $12$: sum = $10$ ✘
- $3$ and $-8$: sum = $-5$ ✘
- $-3$ and $8$: sum = $5$ ✔ ← This works!

So the two numbers are 8 and −3.

3. Now rewrite the middle term $5x$ as $8x - 3x$:
$$
6x^2 + 8x - 3x - 4
$$

4. Group and factor by grouping:
$$
(6x^2 + 8x) + (-3x - 4)
= 2x(3x + 4) -1(3x + 4)
= (2x - 1)(3x + 4)
$$

Alternatively, using the formula shown in the image:
$$
\frac{1}{a}(ax + m)(ax + n)
$$
where $m = 8$, $n = -3$, and $a = 6$:
$$
\frac{1}{6}(6x + 8)(6x - 3)
$$

Now simplify each binomial by factoring out common factors:

- $6x + 8 = 2(3x + 4)$
- $6x - 3 = 3(2x - 1)$

So:
$$
\frac{1}{6} \cdot 2(3x + 4) \cdot 3(2x - 1) = \frac{1}{6} \cdot 6 \cdot (3x + 4)(2x - 1) = (3x + 4)(2x - 1)
$$

That matches what we got earlier.

✔ Final factored form: $(3x + 4)(2x - 1)$

Final Answer:
(3x + 4)(2x - 1)