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Find all solutions to the equation x^2 + 29 = 10x + 3.

 Jun 25, 2021
 #1
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$x^2+29=10x+3$


take the 3 to the left-hand side


$  x^2+26=10x $


now take the 10x to the other side


$  x^2-10x+26=0 $

 

now to solve this you can pick many ways, though i will pick using the quadratic formula


$ _2x_1=\frac{-b\pm \sqrt{b^2-4ac}}{2a}  $


where $a=1$  ;  $b=-10$ ;  $c=26$

 

just plug and chug

 

$ _2x_1=\frac{-\left(-10\right)\pm \sqrt{\left(-10\right)^2-4\cdot \:1\cdot \:26}}{2\cdot \:1} $

 

$ _2x_1=\frac{-\left(-10\right)\pm \iota \sqrt{10^2-104}} {2\cdot \:1} $

 

$ _2x_1=\frac{-\left(-10\right)\pm \iota \sqrt{4} } {2\cdot \:1} $

 

$ _2x_1=\frac{-\left(-10\right)\pm \:2\iota }{2\cdot \:1} $

 

that gives us $ x_1=\frac{10+2\iota }{2\cdot \:1}  \ \ \ \Rightarrow \ \ \ \frac{10+2\iota }{2}   \ \ \ \Rightarrow \ \ \ \frac{\cancel{2}\left(5+\iota \right)}

 

{\cancel{2}}  $  

 

finally, we get $x_1==5+\iota $

 

second root will be as following:

 

$x_2=\frac{10-2\iota }{2\cdot \:1}  \ \ \ \Rightarrow \ \ \ \frac{10-2\iota a}{2} \ \ \ \Rightarrow \ \ \ \frac{\cancel{2}\left(5-\iota \right)}{\cancel{2}}$

 

thus the second root is $  x_2=5-\iota $


 

 Jun 25, 2021
 #2
avatar+139 
+2

that gives us $ x_1=\frac{10+2\iota }{2\cdot \:1}  \ \ \ \Rightarrow \ \ \ \frac{10+2\iota }{2}   \ \ \ \Rightarrow \ \ \ \frac{\cancel{2}\left(5+\iota \right)}{ \cancel{2}}  $  

 

finally, we get 

 

$  x_1=5+\iota$

 

fixed.

UsernameTooShort  Jun 25, 2021

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