15. (a) Factorise $3x + 6y$ - Junior Cycle Mathematics - Question 15 - 2018

To multiply out the expression $(2x + 7)(x - 4)$, we apply the distributive property:

1. First, distribute $2x$:

   • $2x * x = 2x^2$
   • $2x * (-4) = -8x$

2. Next, distribute $7$:

   • $7 * x = 7x$
   • $7 * (-4) = -28$

3. Now, combine the results: $2x^2 - 8x + 7x - 28$ $= 2x^2 - x - 28$

To solve the quadratic equation $x^2 + 6x - 16 = 0$, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$, $b = 6$, and $c = -16$. Plugging the values into the formula:

1. Calculate the discriminant:

   • $b^2 - 4ac = 6^2 - 4(1)(-16) = 36 + 64 = 100$

2. Substitute into the quadratic formula: $x = \frac{-6 \pm \sqrt{100}}{2(1)} = \frac{-6 \pm 10}{2}$

3. This results in two solutions:

   • $x = \frac{4}{2} = 2$
   • $x = \frac{-16}{2} = -8$

Therefore, the solutions are $x = 2$ or $x = -8$.

To simplify the expression $x^2 + 6x - 16 + (x - 2)$, we first distribute and combine the like terms:

1. Rewrite the expression: $x^2 + 6x - 16 + x - 2$

2. Combine the like terms:

   • $6x + x = 7x$
   • $-16 - 2 = -18$

3. The simplified expression is: $x^2 + 7x - 18$