What are the solutions to: x^2 + 7x + 11 = 11x + 9?

Hi,

x² + 7x + 11 = 11x + 9

x² - 4x + 2 = 0
...........__________
-(-4) ± √(-4)² - 4(1)(2)
----------------------------- = x
.............2(1)

........_____
4 ± √16 - 8
----------------- = x
.............2

........._
4 ± 2√2
----------- = x
....2

......._
2 ± √2 = x <==ANSWER

I hope that helps!! :-)

x² - 4x + 2 = 0
x = [ 4 ± √ (16 - 8 ) ] / 2
x = [ 4 ± √8 ] / 2
x = [ 4 ± 2√2 ] / 2
x = 2 ± √2

Question Number 1 :
For this equation x^2 + 7*x + 11 = 11*x + 9 , answer the following questions :
A. Find the roots using Quadratic Formula !
B. Use completing the square to find the root of the equation !

Answer Number 1 :
First, we have to turn equation : x^2 + 7*x + 11 = 11*x + 9 , into a*x^2+b*x+c=0 form.
x^2 + 7*x + 11 = 11*x + 9 , move everything in the right hand side, to the left hand side of the equation
<=> x^2 + 7*x + 11 - ( 11*x + 9 ) = 0 , which is the same with
<=> x^2 + 7*x + 11 + ( - 11*x - 9 ) =0 , now open the bracket and we get
<=> x^2 - 4*x + 2 = 0

The equation x^2 - 4*x + 2 = 0 is already in a*x^2+b*x+c=0 form.
As the value is already arranged in a*x^2+b*x+c=0 form, we get the value of a = 1, b = -4, c = 2.

1A. Find the roots using Quadratic Formula !
By using abc formula the value of x is both
x1 = (-b+sqrt(b^2-4*a*c))/(2*a) and x2 = (-b-sqrt(b^2-4*a*c))/(2*a)
As a = 1, b = -4 and c = 2,
we just need to subtitute the value of a,b and c in the abc formula.
Which produce x1 = (-(-4) + sqrt( (-4)^2 - 4 * (1)*(2)))/(2*1) and x2 = (-(-4) - sqrt( (-4)^2 - 4 * (1)*(2)))/(2*1)
Which is the same with x1 = ( 4 + sqrt( 16-8))/(2) and x2 = ( 4 - sqrt( 16-8))/(2)
Which is the same as x1 = ( 4 + sqrt( 8))/(2) and x2 = ( 4 - sqrt( 8))/(2)
We can get x1 = ( 4 + 2.82842712474619 )/(2) and x2 = ( 4 - 2.82842712474619 )/(2)
So we have the answers x1 = 3.4142135623731 and x2 = 0.585786437626905

1B. Use completing the square to find the root of the equation !
x^2 - 4*x + 2 = 0 ,divide both side with 1
By doing so we get x^2 - 4*x + 2 = 0 ,
And the coefficient of x is -4
We have to use the fact that ( x + q )^2 = x^2 + 2*q*x + q^2 , and assume that q = -4/2 = -2
Next, we have to separate the constant to form x^2 - 4*x + 4 - 2 = 0
So we will get ( x - 2 )^2 - 2 = 0
Which is the same with (( x - 2 ) - 1.4142135623731 ) * (( x - 2 ) + 1.4142135623731 ) = 0
By using the associative law we get ( x - 2 - 1.4142135623731 ) * ( x - 2 + 1.4142135623731 ) = 0
Just add up the constants in each brackets, and we get ( x - 3.4142135623731 ) * ( x - 0.585786437626905 ) = 0
So we have the answers x1 = 3.4142135623731 and x2 = 0.585786437626905

x^2 + 7x + 11 = 11x + 9
x^2 + 7x - 11x + 11 - 9 = 0
x^2 - 4x + 2 = 0
x = [-b ±√(b^2 - 4ac)]/2a

a = 1
b = -4
c = 2

x = [4 ±√(16 - 8)]/2
x = [4 ±√8]/2
x = [4 ±2.82]/2 (approx.)

x = [4 + 2.82]/2
x = 6.82/2
x = 3.41

x = [4 - 2.82]/2
x = 1.18/2
x = 0.59

∴ x = 0.59 , 3.41

The equation states that x must be either 2+sqrt(2) or 2-sqrt(2). The reasoning is as follows.
The equation means the same thing as x^2+11x-4x+9+2=11x+9 and this means the same as x^2-4x+2=0.
Note that (x-2)^2 is x-4x+4.
Clearly then, since the equation is the same as x^2-4x+4-2=0, it can also be written (x-2)^2-2=0.
In this form, the equation can be solved quite straightforwardly, by applying the inverses of the operations to which x is subject to both sides of the equation in the reverse order. In other words (x-2)^2=2, x-2=sqrt(2) and x=sqrt(2)+1. The equation defines two possible values of x because there are two square roots (sqrt) of a positive number, both of the same magnitude, one positive and one negative.

x^2 + 7x + 11 = 11x + 9
x^2 - 4x + 2 = 0

use quadratic formula

x = [-b +/- (b^2 - 4ac)^1/2 ] / 2a

x = [4 +/- 2(2)^(1/2)] / 2
= 2 +/- (2)^1/2

thus, x = 2 + (2)^(1/2) or x = 2 - (2)^(1/2)

x^2+7x+11=11x+9
x^2-4x+2=0

use the quadratic equation to solve

4(+or-)sqrt(8)/2
4(+or-)2.83/2
2(+or-)1.415
3.415
0.585

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x^2-4x+2=0
x=4+-sqrt8/2=2+-sqrt2