Solve for x ......√(5 - x ) = 5 - x {}^(2)

Question

Solve for x ......
`√(5 - x ) = 5 - x {}^(2)`

Answer 1

Answer:
`\begin{gathered} The\text{ value of x for the question:} \\ \text{x = -1.562 or x = 1.791} \end{gathered}`

Step-by-step explanation:
`\begin{gathered} Given: \\ \sqrt{5\text{ - x}}\text{ = 5 - x}^2 \\ We\text{ need to find the value of x} \end{gathered}`

To answer the question, we will solving and graphing to get the value of x

`\begin{gathered} \sqrt{5\text{ - x}}\text{ = 5 - x}^2 \\ Square\text{ both sides:} \\ (\sqrt{5\text{ - x}})^2\text{ = \lparen5 - x}^2)^2 \\ 5\text{ - x = \lparen5 - x}^2)(5\text{ - x}^2)\text{ } \end{gathered}`
`\begin{gathered} 5\text{ - x = 5\lparen5 - x}^2)\text{ - x}^2(5\text{ - x}^2) \\ 5\text{ - x = 25 - 5x}^2\text{ - 5x}^2\text{ + x}^4 \\ 5\text{ - x = 25 - 10x}^2\text{ + x}^4 \\ x^4\text{ - 10x}^2\text{ + x + 25 - 5 = 0} \\ x^4\text{ -10x}^2\text{ + x + 20 = = 0} \end{gathered}`
`\begin{gathered} -x\text{ = x}^4\text{ - 10x}^2\text{ + 20} \\ \\ The\text{ highest degree in our expansion above is 4} \\ So\text{ to solve for x, we will represent x}^2\text{ with u} \\ \\ The\text{ expression becomes:} \\ u^2\text{ - 10u + 20 = -x} \\ u^2\text{ - 10 u + 20} \\ u\text{ = }\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ u\text{ = }(-(-10)\pm√((-10)^2-4(1)(20)))/(2(1)) \\ \text{ u = }(10\pm√(100-80))/(2) \\ u\text{ = }(10\pm2√(5))/(2)\text{ = }5\pm√(5) \\ u\text{ = 7.2361 or 2.7639} \end{gathered}`

`\begin{gathered} NB:I\text{ couldn't proceed further because we have x in the expression after expansion.} \\ When\text{ the degree of the polynomial is to the 4th degree, and we want to solve algebraically,} \\ we\text{ need to have just 3 terms \lparen4th power, 2nd power and a constant\rparen to be able to represent } \\ \text{with a letter. }I\text{ n this case letter u} \end{gathered}`
`\begin{gathered} left\text{ hand side will be equated to y} \\ and\text{ te right side equated to y} \\ Plotting\text{ both graphs on same coordinate} \\ The\text{ point of intersection of both graph will be the solution} \\ \\ From\text{ our graph:} \\ The\text{ point of intersection: }(-1.562,\text{ 2.562\rparen and \lparen1.791, 1.791\rparen} \end{gathered}`
`\begin{gathered} The\text{ value of x for the question:} \\ \text{x = -1.562 or x = 1.791} \end{gathered}`