solve for x \sqrt{x^2-4x+8} +x=2 - x please show all workings​

Question

solve for x
\sqrt{x^2-4x+8} +x=2 - x
please show all workings​

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Thái Dương 4 years 2021-07-30T08:24:52+00:00 2 Answers 22 views 0

Answers ( )

  1. Latifah
    0
    2021-07-30T08:26:19+00:00
    Reply

    Step-by-step 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!

  2. mocmien2
    0
    2021-07-30T08:26:31+00:00
    Reply

    Answer:

    \displaystyle x=-\frac{2}{3}

    Step-by-step explanation:

    We want to solve the equation:

    \displaystyle \sqrt{x^2-4x+8}+x=2-x

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

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

    And square both sides:

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

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

    Testing x = 2:

    \displaystyle \begin{aligned}\sqrt{(2)^2-4(2)+8}+(2) &\stackrel{?}{=}2-(2) \\ \\ \sqrt{4-8+8}+2&\stackrel{?}{=}0 \\ \\ \sqrt{4}+2&\stackrel{?}{=}0 \\ \\ 2+2=4&\neq 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.  

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