Question

How do you solve the equation by completing the square `x^2-14x+33=0`?

Answer 1

`x = 11` or `x = 3`.

Explanation:

The idea of completing the square, otherwise abbreviated as CTS, is not necessarily complex, but it's a bit involved. First, let's remember that a quadratic equation is of the following format:
`ax^2 + bx + c = 0`, where `a, b` and `c` are coefficients of the equation. Coefficient is just a fancy word for numbers attached to the variable (in this case, the variable x). For this equation, we can easily identify our `a, b` and `c` numbers! Just eyeball it.

`a = 1`, since `x^2 = 1 \cdot x^2`.
`b = -14`
`c = 33`

Now that we've got that out of the way, let's consider the following algebraic term: `r = (\frac{b}{2a})^2`. Our goal is now to calculate `r`. I'll explain what this means a bit later on, but if I do so now, you'll get bogged down.

So, `r = (\frac{b}{2a})^2 = (\frac{-14}{2\cdot 1})^2 = (-7)^2 = 49`

Great. But, now what? Well, we add 49 to both sides of the equation! Analyze how I add it in, though.

`(x^2 -14x + 49) + 33 = 49`.

See what I did there? I injected 49 conveniently, right next to -14x. The question you should be asking is, `why?` Well, this is factorable! In fact, it factors into the following:

`(x - 7)^2 + 33 = 49`
Notice how we used the term `\frac{b}{2a} = \frac{-14}{2 \cdot 1} = -7` to find the factored form.

Now, let's subtract 33 from both sides.
`(x - 7)^2 = 16`

Now, this is the easy-peezy-lemon-squeazy part. Take the square root of both sides. In doing so, you'll get rid of the exponential 2 on the left side of your equation and on the right side, you'll take the square root of 16. So,

`\sqrt{(x-7)^2} = \sqrt{16}`
simplifies to
`x - 7 = \pm 4`

Which further simplifies to

`x = 7 \pm 4`.

Your solutions are: `x_{1} = 7+4 = 11` or `x_{2} = 7 - 4 = 3`. That's about it.

Explanation of the `r = (\frac{b}{2a})^2` term:
This comes from the proof of the quadratic formula. In fact, one of the ways in which we prove the quadratic formula is through the CTS methodology.