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To answer the question, we will solving and graphing to get the value of x
[tex]\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}[/tex][tex]\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}[/tex][tex]\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{ = }\frac{-(-10)\pm\sqrt{(-10)^2-4(1)(20)}}{2(1)} \\ \text{ u = }\frac{10\pm\sqrt{100-80}}{2} \\ u\text{ = }\frac{10\pm2\sqrt{5}}{2}\text{ = }5\pm\sqrt{5} \\ u\text{ = 7.2361 or 2.7639} \end{gathered}[/tex][tex]\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}[/tex][tex]\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}[/tex][tex]\begin{gathered} The\text{ value of x for the question:} \\ \text{x = -1.562 or x = 1.791} \end{gathered}[/tex]
