Video: Solving Equations Involving Root Functions

For what numbers 𝑐 can we solve √(𝑥 − 7) = 𝑐?

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Video Transcript

For what numbers 𝑐 can we solve the square root of 𝑥 minus seven is equal to 𝑐?

In order to calculate the possible values of 𝑐, we firstly need to work out what values of 𝑥 we can substitute into the left-hand side of the equation. We know that in order to get a real solution, we can only square root numbers that are greater than or equal to zero. This means that 𝑥 minus seven must be greater than or equal to zero. Adding seven to both sides of this equation gives us 𝑥 is greater than or equal to seven. When 𝑥 is equal to seven, 𝑐 is equal to the square root of seven minus seven. This is equal to zero. As 𝑥 minus seven must be greater than or equal to zero, which means that 𝑥 is greater than or equal to seven, then 𝑐 must be greater than or equal to zero.

Another way of thinking about this problem would be to rearrange our initial equation. Squaring both sides of the equation, we get 𝑥 minus seven is equal to 𝑐 squared. We can then add seven to both sides of this equation so that 𝑥 is equal to 𝑐 squared plus seven. If 𝑐 is greater than or equal to zero, we can find the value of 𝑥 we need to input into the equation as 𝑥 is equal to 𝑐 squared plus seven.

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