Question Video: Solving Equations Involving Root Functions Mathematics • 10th Grade

Solve √(𝑥 − 7) = −3.


Video Transcript

Solve the square root of 𝑥 minus seven equals negative three.

When we’re working with equations like these involving square roots, we usually solve them algebraically. However, any time we’re working with square roots or any other type of roots, we have to be aware that we could create extraneous solutions. This means we could find a value for 𝑥 that doesn’t make this statement true.

On the left, we have a square root of 𝑥 minus seven. And on the right, the value negative three is a negative integer. This is a real number that is less than zero, but the square root of 𝑥 minus seven can never produce a negative integer because an integer is a real number. And if we have a value inside the radical that is negative, it will produce imaginary solutions. And so we can say that the square root of 𝑥 minus seven will never be equal to negative three.

By inspecting this equation closely, we can say that this has no solution. But what if you didn’t recognize this and tried to solve algebraically? You would have the square root of 𝑥 minus seven equals negative three. We then square both sides of the equation, which gives us 𝑥 minus seven equals nine. If we add seven to both sides, we get a statement that says 𝑥 equals 16. We’ve already said there’s no possible solution. And this is why if you were solving this algebraically, you would need to check to make sure that 𝑥 equals 16 is a valid solution.

If we try to plug in 𝑥 equals 16 into this equation, we get the square root of 16 minus seven equals negative three, that the square root of nine equals negative three. But the square root of nine is three. And three is not equal to negative three, which means 𝑥 cannot be equal to 16. It is an extraneous solution. And this equation has no solution.

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