Video: Solving Quadratic Equations by Completing the Square

By completing the square, write all the solutions to π‘₯Β² βˆ’ 4π‘₯ + 1 = 0.

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Video Transcript

By completing the square, write all the solutions to π‘₯ squared minus four π‘₯ plus one equals zero.

So if we take a look at the equation in our question, we can see that it’s in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐, where π‘Ž is the coefficient of π‘₯ squared, 𝑏 is the coefficient of π‘₯, and 𝑐 is the final integer and then this is all equal to zero.

So when we have an equation in this form with the single coefficient of π‘₯ squared, then we actually have a general form for completing the square. And that form involves the π‘₯ squared and the π‘₯ terms. So we can say that if we have π‘₯ squared plus 𝑏π‘₯, then this is equal to π‘₯ plus 𝑏 over two. So we halve the coefficient of π‘₯ all squared minus the coefficient of π‘₯ again halved, so 𝑏 over two all squared. And then, what we’d actually do in our situation is then add on our 𝑐.

So now we have the general form, what we can do is use this to complete the square and find the solutions to our equation. So when taking a look at our equation, we can see that 𝑏 is equal to negative four. And that’s because the coefficient of π‘₯ is negative four. So therefore, completing the square, what we have is π‘₯ minus four over two all squared. And we get π‘₯ minus four over two because if we have negative four as our 𝑏, so our coefficient of π‘₯, then if we add a negative, it’s just the same as subtract.

So that’s why we have π‘₯ minus four over two all squared minus negative four over two all squared plus one equals zero which gives us π‘₯ minus two all squared minus four plus one is equal to zero, which gives us the simplified completed square form of π‘₯ minus two all squared minus three equals zero.

So this is the first part of the question answered because we’ve completed the square. Now, what we need to do is find all of the solutions to π‘₯ squared minus four π‘₯ plus one is equal to zero.

So now, the first stage to find the solutions for π‘₯ is to add three to each side of our equation. And when we do that, we get π‘₯ minus two all squared is equal to three. Then, the next stage is to take the square root of each side of the equation. So this is gonna give us π‘₯ minus two is equal to positive or negative root three.

A quick tip here: make sure you include this because if we don’t have both positive and negative values, then we’ll miss one of our solutions because there will be two possible solutions.

So then, the next step is to add two to each side of the equation. And when we do that, we get π‘₯ is equal to two plus or minus root three.

So therefore, we can say that by completing the square, all the solutions to π‘₯ squared minus four π‘₯ plus one equal zero are π‘₯ is equal to two plus root three or π‘₯ is equal to two minus root three.

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