The extraneous solutions for the Avery equation is [tex]z = \frac{7}{3}[/tex].
What is an equation?
The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an ‘equal’ sign. The most basic and simple algebraic equations consist of one or more variables in math.
Given equation
[tex]\sqrt{z^{2}+8 } =1-2z[/tex]
Squaring on both sides
[tex]z^{2} +8=(1-2z)^{2}[/tex]
[tex](1-2z)^{2}[/tex] is in the form of [tex](a-b)^{2} =a^{2} -2ab+b^{2}[/tex].
[tex]z^{2}+8=1-4z+4z^{2}[/tex]
[tex]4z^{2} -z^{2} -4z+1-8=0[/tex]
[tex]3z^{2} -4z-7=0[/tex]
[tex]3z^{2} +3z-7z-7=0[/tex]
[tex]3z(z+1)-7(z+1)=0[/tex]
[tex](3z-7)(z+1)=0[/tex]
[tex]z=\frac{7}{3}[/tex] or [tex]z=-1[/tex]
because [tex]\sqrt{z^{2}+8 } \ge 0[/tex] so [tex]1-2z \ge 0[/tex]
When [tex]z=\frac{7}{3}[/tex] [tex]1-2 \times \frac{7}{3} =-\frac{11}{3} < 0[/tex](Abandon)
When z = -1 1 - 2 × (-1) = 1 + 2 = 3 > 0
So the extraneous solutions is [tex]z = \frac{7}{3}[/tex].
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