# How do you solve  sqrt(-10x - 4) = 2x?

##### 2 Answers
Apr 10, 2015

This has no answer. At first glance:

1. Square it (including every sign outside the square root).
2. Bring everything else to the other side with the 2nd degree term.

You get:
$4 {x}^{2} + 10 x + 4 = 0$

Divide by 2.
$2 {x}^{2} + 5 x + 2 = 0$

Factor.
$\left(2 x + 1\right) \left(x + 2\right) = 0$

$x = - \frac{1}{2} , x = - 2$

Check:

$\sqrt{- 10 \left(- \frac{1}{2}\right) - 4} = 2 \left(- \frac{1}{2}\right) \implies$ 1 is not equal to -1. This answer is extraneous, even though -1 is equal to -1 by saying the square root yields the positive and negative answer. One of them is false, so the result is contingently true.

$\sqrt{- 10 \left(- 2\right) - 4} = 2 \left(- 2\right) \implies$ 4 is not equal to -4. So is this one, even though -4 is equal to -4 by saying the square root yields the positive and negative answer. One of them is false, so the result is contingently true.

As-written, there are no solutions. Math at its core is pure logic that is supposed to always be true if done correctly, not just sometimes . There has to be a typo on the sign on the left, or it's a trick question. It should be:

$- \sqrt{- 10 x - 4} = 2 x$

if there is to be an answer. This is a result of isolating one result of a square root. It has to be a $\pm \sqrt{s t u f f}$ result, and only one of them had a real solution.

Apr 10, 2015

This equation has no solutions.

Though there could be a typo and it could be:

$- \sqrt{- 10 x - 4} = 2 x$

We can square both sides to get

$- 10 x - 4 = {\left(2 x\right)}^{2}$

$- 10 x - 4 = 4 {x}^{2}$

This gives us a quadratic equation:

$4 {x}^{2} + 10 x + 4 = 0$

Dividing both sides by 2, we get:

$\frac{4 {x}^{2} + 10 x + 4}{2} = \frac{0}{2}$

$2 {x}^{2} + 5 x + 2 = 0$

We use the Splitting the Middle Term technique to factorise the expression on the left

$2 {x}^{2} + 4 x + x + 2 = 0$

$2 x \cdot \left(x + 2\right) + 1 \cdot \left(x + 2\right) = 0$

$\left(2 x + 1\right) \cdot \left(x + 2\right) = 0$

This tells us that

$2 x + 1 = 0$ or $x + 2 = 0$

color(green)(x = -1/2  or color(green)( x = -2

x can take either of these values and both will satisfy the equation $- \sqrt{- 10 x - 4} = 2 x$