Question

Solve for x

`√(x^2-4x+8) +x=2 - x`
please show all workings​

Answer 1

Explanation:

Hey there!

Given;

`\sqrt{ {x}^(2) - 4x + 8} + x = 2 - x`

Take "X" in right side.

`\sqrt{ {x - 4 + 8}^(2) } = 2 - 2x`

Squaring on both sides;

`{( \sqrt{ {x}^(2) - 4x + 8 } )}^(2) = {(2 - 2x)}^(2)`

Simplify;

`{x}^(2) - 4x + 8 = {(2)}^(2) - 2.2.2x + {(2x)}^(2)`

`{x }^(2) - 4x + 8 = 4 - 8x + 4 {x}^(2)`

`3 {x}^(2) - 4x - 4 = 0`

`3 {x}^(2) - (6 - 2)x - 4 = 0`

`3 {x}^(2) - 6x + 2x - 4 = 0`

`3x(x - 2) + 2(x - 2) = 0`

`(3x + 2)(x - 2) = 0`

Either;

3x+2 = 0

x= -2/3

Or;

x-2 = 0

x= 2

Check:

Keeping X= -2/3,

√(x²-4x+8 ) +X = 2-x

√{(-2/3)²-4*-2/3+8}+(-2/3) = 2+2/3

8/3 = 8/3 (True)

Now; Keeping X= 2

√{(2)²-4*2+8}+2 = 2-2

8 ≠0 (False)

Therefore, the value of X is -2/3.

Hope it helps!

Answer 2

`\displaystyle x=-(2)/(3)`

Explanation:

We want to solve the equation:

`\displaystyle √(x^2-4x+8)+x=2-x`

We can isolate the square root. Subtract x from both sides:

`√(x^2-4x+8)=2-2x`

And square both sides:

`(√(x^2-4x+8))^2=(2-2x)^2`

Expand:

`x^2-4x+8=4-8x+4x^2`

Isolate the equation:

`3x^2-4x-4=0`

Factor:

`\displaystyle (3x+2)(x-2)=0`

Zero Product Property:

`3x+2=0\text{ or } x-2=0`

Solve for each case. Hence:

`\displaystyle x=-(2)/(3)\text{ or } x=2`

Now, we need to check for extraneous solutions. To do so, we can substitute each value back into the original equation and check whether or not the resulting statement is true.

Testing x = -2/3:

`\displaystyle \begin{aligned} \sqrt{\left(-(2)/(3)\right)^2-4\left(-(2)/(3)\right)+8}+\left(-(2)/(3)\right)&\stackrel{?}{=}2-\left(-(2)/(3)\right)\\ \\ \sqrt{(4)/(9)+(8)/(3)+8}-(2)/(3)&\stackrel{?}{=}2+(2)/(3) \\ \\ \sqrt{(100)/(9)}-(2)/(3)& \stackrel{?}{=} (8)/(3)\\ \\ (10)/(3)-(2)/(3) =(8)/(3)& \stackrel{\checkmark}{=}(8)/(3)\end{aligned}`

Since the resulting statement is true, x = -2/3 is indeed a solution.

Testing x = 2:

`\displaystyle \begin{aligned}√((2)^2-4(2)+8)+(2) &\stackrel{?}{=}2-(2) \\ \\ √(4-8+8)+2&\stackrel{?}{=}0 \\ \\ √(4)+2&\stackrel{?}{=}0 \\ \\ 2+2=4&\\eq 0\end{aligned}`

Since the resulting statement is not true, x = 2 is not a solution.

Therefore, our only solution to the equation is x = -2/3.