# How do you solve y^2 + 10y + 35 = 0 by completing the square?

Aug 3, 2016

$y = - 5 + \sqrt{- 10}$
$y = - 5 - \sqrt{- 10}$

#### Explanation:

${y}^{2} + 10 y + 35 = 0$
or
${y}^{2} + 10 y + 25 + 10 = 0$
or
${y}^{2} + 2 \left(y\right) 5 + {5}^{2} + 10 = 0$
or
${\left(y + 5\right)}^{2} = - 10$
or
$y + 5 = \pm \sqrt{- 10}$
or
$y = - 5 \pm \sqrt{- 10}$
or
$y = - 5 + \sqrt{- 10}$========Ans $1$
or
$y = - 5 - \sqrt{- 10}$========Ans $2$

Aug 3, 2016

$y = - 5 \pm \left(\sqrt{- 10}\right) i$

#### Explanation:

This is a quadratic in $y$ instead of in $x$. This means that the plot will be rotated ${90}^{0}$ and be of shape $\subset$ or $\supset$.

Set as $\text{ } x = {y}^{2} + 10 y + 35$

$\textcolor{b l u e}{\text{Step 1}}$

Write as:$\text{ } x = \left({y}^{2} + 10 y\right) + 35 + k$

The $k$ will be needed to correct an error that this approach introduces.
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$\textcolor{b l u e}{\text{Step 2}}$

Take the index (power) outside the brackets
$x = {\left(y + 10 y\right)}^{2} + 35 + k$

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$\textcolor{b l u e}{\text{Step 3}}$

Discard the $y$ from $10 y$
$x = {\left(y + 10\right)}^{2} + 35 + k$

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$\textcolor{b l u e}{\text{Step 4}}$
Halve the 10
$x = {\left(y + 5\right)}^{2} + 35 + k$

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$\textcolor{b l u e}{\text{Step 5}}$
Now we need to correct for the error. The 5 in ${\left(y + 5\right)}^{2}$ produces ${5}^{2}$ which is a value that is not in the original equation

So ${5}^{2} + k = 0 \implies k = - 25$ giving:

color(brown)(x=(y+5)^2+35+k)color(blue)(" "->" "x=(y+5)^2+35-25

$\textcolor{g r e e n}{x = {\left(y + 5\right)}^{2} + 10} \text{ "larr" Completed the square}$

This is also known as the Vertex Form Equation
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$\textcolor{b l u e}{\text{Solving for } {\left(y + 5\right)}^{2} + 10 = 0}$

${\left(y + 5\right)}^{2} = - 10$

Square root both sides

$y + 5 = \pm \sqrt{- 10}$

As soon as you see a root of a negative it means Complex Numbers are involved. In other words; the plot does not cross the y-axis in this case. If the graph was of general shape $\cup \text{ or } \cap$ it would not cross the x-axis. This graph is of shape $\subset$.

$y = - 5 \pm \left(\sqrt{- 10}\right) i$