# How do you factor 3x ^ { 2} + 8x - 528= 0?

Sep 1, 2017

$x = 12$ or $x = - \frac{44}{3}$

#### Explanation:

To factor $a {x}^{2} + b x + c = 0$, we should split the middle term $b$ in two parts, so that the product of two parts is $a c$.

In $3 {x}^{2} + 8 x - 528 = 0$, we should split the middle term $8$ in two parts, so that the product of two parts is $3 \times - 528 = 1584$.

As product is negative, the factors of $- 1584$ will be opposite in sign and as their sum is $8$, the difference in absolute value is $8$ and positive factor is greater.

Factors of $1584$ are $\left(2 , 792\right) , \left(3 , 528\right) , \left(4 , 396\right) , \left(6 , 264\right) , \left(8 , 198\right) , \left(9 , 176\right) , \left(11 , 144\right) , \left(12 , 132\right) , \left(16 , 99\right) , \left(18 , 88\right) , \left(22 , 72\right) , \left(24 , 66\right) , \left(33 , 48\right) , \left(36 , 44\right)$

Now $36$ and $44$ have a difference of $8$

Hence, $3 {x}^{2} + 8 x - 528 = 0$ can bewritten as

$3 {x}^{2} + 44 x - 36 x - 528 = 0$

or $x \left(3 x + 44\right) - 12 \left(3 x + 44\right) = 0$

or $\left(x - 12\right) \left(3 x + 44\right) = 0$

i.e. $x = 12$ or $x = - \frac{44}{3}$

Sep 1, 2017

$\left(x - 12\right) \left(3 x + 44\right) = 0$

#### Explanation:

This can be factored by completing the square.

Given:

$f \left(x\right) = 3 {x}^{2} + 8 x - 528$

To make the arithmetic less messy, we can premultiply by $3$ to make the leading coefficient into a perfect square, then divide by $3$ at the end.

We will also use the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \left(3 x + 4\right)$ and $b = 40$ as follows:

$3 f \left(x\right) = 3 \left(3 {x}^{2} + 8 x - 528\right)$

$\textcolor{w h i t e}{3 f \left(x\right)} = 9 {x}^{2} + 24 x - 1584$

$\textcolor{w h i t e}{3 f \left(x\right)} = {\left(3 x\right)}^{2} + 2 \left(3 x\right) \left(4\right) + 16 - 1600$

$\textcolor{w h i t e}{3 f \left(x\right)} = {\left(3 x + 4\right)}^{2} - {40}^{2}$

$\textcolor{w h i t e}{3 f \left(x\right)} = \left(\left(3 x + 4\right) - 40\right) \left(\left(3 x + 4\right) + 40\right)$

$\textcolor{w h i t e}{3 f \left(x\right)} = \left(3 x - 36\right) \left(3 x + 44\right)$

$\textcolor{w h i t e}{3 f \left(x\right)} = 3 \left(x - 12\right) \left(3 x + 44\right)$

Dividing both ends by $3$ we find:

$3 {x}^{2} + 8 x - 528 = \left(x - 12\right) \left(3 x + 44\right)$

So the given equation can be written:

$\left(x - 12\right) \left(3 x + 44\right) = 0$

which has zeros $x = 12$ and $x = - \frac{44}{3}$

Sep 1, 2017

$\left(3 x + 44\right) \left(x - 12\right) = 0$

$x = - \frac{44}{3} \mathmr{and} x = 12$

#### Explanation:

The first step is to determine whether there are factors.

$a = 3 \text{ } b = 8 \mathmr{and} c = - 528$

${b}^{2} - 4 a c \text{ } \rightarrow {8}^{2} - 4 \left(3\right) \left(- 528\right) = 6400$

$6400$ is a perfect square, therefore there are rational factors.

We need to find factors of $3 \mathmr{and} 528$ whose products differ by $8$.

The smaller the value of $b$, the closer it is to $\sqrt{a c}$

$a c = 3 \times 528 = 1584$

$\sqrt{1584} = 39.799 \approx 40$

Use some trial and error starting from $40$

$40$ is not a factor of $1584$
$41$ is not a factor of $1584$
$42$ is not a factor of $1584$
$43$ is not a factor of $1584$

$\textcolor{red}{44 \times 36 = 1584 \mathmr{and} 44 - 36 = 8} \text{ } \leftarrow$ BINGO!

Now combine factors of $3 \mathmr{and} 528$ to get to $44 \mathmr{and} 36$

$\text{ } 3 \mathmr{and} 528$
$\text{ } \downarrow \textcolor{w h i t e}{\times \times} \downarrow$
$\text{ "3color(white)(xxxxx)44" } \rightarrow 1 \times 44 = 44$
$\text{ "1color(white)(xxxxx)12" } \rightarrow 3 \times 12 = \underline{36}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times} 8$

We have the factors, now add in the signs to get $+ 8$

$\text{ "3 and" } 528$
$\text{ } \downarrow \textcolor{w h i t e}{\times \times x} \downarrow$
$\text{ "3color(white)(xxxxx)+44" } \rightarrow 1 \times 44 = + 44$
$\text{ "1color(white)(xxxxx)-12" } \rightarrow 3 \times 12 = - \underline{36}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times} + 8$

$\left(3 x + 44\right) \left(x - 12\right) = 0$

If we use the factors to solve the equation we get:

$3 x + 44 = 0 \text{ } \rightarrow x = - \frac{44}{3}$

$x - 12 = 0 \text{ } \rightarrow x = 12$