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

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

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Answer 1 · 2016-08-03T17:13:48

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

Explanation:

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

Answer 2 · 2016-08-03T18:22:19

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

Explanation:

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

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

$Step 1$ $color(blue)("Step 1")$

Write as: $x = (y^{2} + 10 y) + 35 + k$ $" "x=(y^2+10y)+35+k$

The $k$ $k$ will be needed to correct an error that this approach introduces.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$Step 2$ $color(blue)("Step 2")$

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

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Step 3$ $color(blue)("Step 3")$

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

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Step 4$ $color(blue)("Step 4")$
Halve the 10
$x = {(y + 5)}^{2} + 35 + k$ $x=(y+5)^2+35+k$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Step 5$ $color(blue)("Step 5")$
Now we need to correct for the error. The 5 in ${(y + 5)}^{2}$ $(y+5)^2$ produces $5^{2}$ $5^2$ which is a value that is not in the original equation

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

$x = {(y + 5)}^{2} + 35 + k \to x = {(y + 5)}^{2} + 35 - 25$ $color(brown)(x=(y+5)^2+35+k)color(blue)(" "->" "x=(y+5)^2+35-25$

$x = {(y + 5)}^{2} + 10 \leftarrow Completed the square$ $color(green)(x=(y+5)^2+10) " "larr" Completed the square"$

This is also known as the Vertex Form Equation
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Solving for {(y + 5)}^{2} + 10 = 0$ $color(blue)("Solving for "(y+5)^2+10=0)$

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

Square root both sides

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

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 $\cup or \cap$ $uu" or "nn$ it would not cross the x-axis. This graph is of shape $\subset$ $sub$ .

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

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

2 Answers

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How do you solve y^2 + 10y + 35 = 0y2+10y+35=0y^2 + 10y + 35 = 0 by completing the square?

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