How do extraneous solutions arise?

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How do extraneous solutions arise?

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Answer 1 · 2015-03-08T04:12:44

In general, extraneous solutions arise when we perform non-invertible operations on both sides of an equation. (That is, they sometimes arise, but not always.)

Non-invertible operations include: raising to an even power (odd powers are invertible), multiplying by zero, and combining sums and differences of logarithms.

Example :
The equations: $x + 2 = 9$ $x+2=9$ and $x = 7$ $x=7$ , have exactly the same set of solutions. Namely: ${7}$ ${7}$ .

Square both sides of $x = 7$ $x=7$ to get the new equation: $x^{2} = 49$ $x^2=49$ . The solution set of this new equation is; ${- 7, 7}$ ${-7, 7}$ . The $- 7$ $-7$ is an extraneous solution introduced by squaring the two expressions

Square both sides of $x + 2 = 9$ $x+2=9$ to get the new equation: $x^{2} + 4 x + 4 = 81$ $x^2+4x+4=81$ . Solve the new equation:
$x^{2} + 4 x - 77 = 0$ $x^2+4x-77=0$ so $(x - 7) (x + 11) = 0$ $(x-7)(x+11)=0$ whose solution set is ${7, - 11}$ ${7,- 11}$ . The $- 11$ $-11$ does not solve the original equation.

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How do extraneous solutions arise?

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