Question Video: Solving Quadratic Equations by Inspection | Nagwa Question Video: Solving Quadratic Equations by Inspection | Nagwa

Question Video: Solving Quadratic Equations by Inspection Mathematics

Solve the equation (𝑥 − 2)² = 7.

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Video Transcript

Solve the equation 𝑥 minus two squared equals seven.

The word solve here tells us we want to find out the 𝑥-values that makes this equation true. To solve for 𝑥, we want to get 𝑥 by itself. And that means we need to consider this equation 𝑥 minus two squared equals seven. We first need to take into account that we are squaring 𝑥 minus two. And if we take the square root of 𝑥 minus two squared, we would just be left with 𝑥 minus two. But if we take the square root of one side of the equation, we need to take the square root of the other side of the equation. And this is where we need to be careful.

We need to consider both square roots of seven, that is, the positive square root of seven and the negative square root of seven. For example, if we take the square root of nine, we’ll either have positive three or negative three because three squared equals nine and negative three squared also equals nine. Which means we now have the statement 𝑥 minus two equals plus or minus the square root of seven. From there, to isolate 𝑥, we’ll add two to both sides of the equation. On the left we’ll have 𝑥, and on the right we’ll have plus or minus the square root of seven plus two.

And at this point, it’s a good idea to break these up into two separate solutions, which is 𝑥 is equal to the positive square root of seven plus two and 𝑥 is equal to the negative square root of seven plus two. We can simplify this a bit further by saying 𝑥 equals two plus the square root of seven, and 𝑥 equals two minus the square root of seven. Again, what we’re saying here is for the value 𝑥 equals two plus the square root of seven and for the value 𝑥 equals two minus the square root of seven, the equation we started with would be true.

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