Solving Equations with Radicals

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In this section, we learn about solving equations with radicals. In order to solve radical equations, we learn about a new property of equality: the squaring property of equality. This squaring property of equality tells us that if each side of the equation is squared, all solutions of the original equation will work in the new equation, but not vice versa. This means we must check all solutions of the transformed equation. Some of the solutions could be extraneous, meaning they will not satisfy the original equation. This comes from the fact that squaring or raising both sides of an equation to any even power causes a loss of information. As an example, suppose we see: 2x = 4. We all know that here x = 2, and only 2. If we square both sides, we would run into an equation that has solutions: x = 2, or x = -2. In this case, x = -2 is extraneous. It is not a solution to the original equation. This is why we must always check to make sure each proposed solution is not extraneous, by plugging into the original equation.
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