# What is x if (4x+3)^2=7?

Mar 15, 2018

${\left(4 x + 3\right)}^{2} = 7$

$16 {x}^{2} + 24 x + 9 - 7 = 0$

$16 {x}^{2} + 24 x + 2 = 0$

$8 {x}^{2} + 12 x + 1 = 0$

$= \frac{\left(- 12\right) \pm \sqrt{\left({12}^{2}\right) - 4 \left(8 \cdot 1\right)}}{2 \cdot 8}$

As the quadratic formula says, from here on I leave you

Mar 15, 2018

$x = \frac{- 3 \pm \sqrt{7}}{\text{4}}$

#### Explanation:

First we expand the left hand side of the equation, by multiplying
$\left(4 x + 3\right) \left(4 x + 3\right) = 7$

Giving us
$16 {x}^{2} + 12 x + 12 x + 9 = 7$

Now we get all terms to one side and combine
$16 {x}^{2} + 24 x + 2 = 0$

Now we can divide all by the constant $2$, giving us
$2 \left(8 {x}^{2} + 12 x + 1\right) = 0$

Now we can divide $2$ on both sides, should look like this
$\frac{2}{2} \left(8 {x}^{2} + 12 x + 1\right) = \frac{0}{2}$

Which simplifies to
$8 {x}^{2} + 12 x + 1 = 0$

Now this does not factor, we have to use the Quadratic Formula which is,
$\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{\text{2a}}$

Where
$a = 8$
$b = 12$
$c = 1$

Now we just plug it in
$\frac{- \left(12\right) \pm \sqrt{{12}^{2} - 4 \left(8\right) \left(1\right)}}{\text{2(8)}}$

You can split this up like so
$- \frac{12}{\text{2(8)"+-(sqrt(12^2-4(8)(1)))/"2(8)}}$

After multipying and combining like terms we get
$- \frac{12}{16} \pm \frac{\sqrt{144 - 32}}{\text{16}}$

This then becomes
$- \frac{3}{4} \pm \frac{\sqrt{112}}{16}$

$- \frac{3}{4} \pm \frac{\sqrt{16} \cdot \sqrt{7}}{16}$

$- \frac{3}{4} \pm \frac{4 \cdot \sqrt{7}}{16}$

$- \frac{3}{4} \pm \frac{\sqrt{7}}{4}$

They have common denominators so we can have this
$x = \frac{- 3 \pm \sqrt{7}}{4}$

Mar 15, 2018

$x = \frac{- 3 \pm \sqrt{7}}{4}$

#### Explanation:

We have: ${\left(4 x + 3\right)}^{2} = 7$

This problem can be done without the use of the quadratic formula.

Let's take the square root of both sides of the equation:

$R i g h t a r r o w 4 x + 3 = \pm \sqrt{7}$

Subtracting $3$ from both sides:

$R i g h t a r r o w 4 x = - 3 \pm \sqrt{7}$

Dividing both sides by $4$:

$\therefore x = \frac{- 3 \pm \sqrt{7}}{4}$