What number is needed to complete the square x2 plus 26x equals 0?

Answer 1

I'm hearing x2 + 26x + ? = 0 where you want to know what the ? would be for a perfect square binomial.

I would be remiss if I did not mention that this is not the best method for solving this equation. First, the best method:

x2 + 26x = 0

First, just factor the x from the left side:

x(x + 26) = 0

Then x = 0 or -26. Problem solved, two answers.

If you really want to complete the square, you can still solve it, but it takes a bit longer. Take half of the middle number (26) which is 13 and then square it. 132=169. That is your question mark. Don't forget to add it to the 0 to keep the equation balanced:

x2 + 26x + 169 = 0 +169

To finish solving:

(x + 13)2 = 169

x + 13 = +/-13

x = -13-13 or x = 13-13

x = -26 or x = 0. Same answer as above. A heckuvalot more complicated.

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While all of the above is true, completing the squares is a very powerful way of solving general quadratic equations. It is implicitly the same as using the quadratic formula.

The question can arise when you want to solve an equation such as

x2 + 26x + 25 = 0

The first step is to rewrite it as

x2 + 26x = -25

Now complete the square on the left, by adding 169 to both sides.

x2 + 26x + 169 = -25 + 169 x2 + 26x + 169 = 144

(x + 13)2 = 12

Take square roots

x + 13 = ± 12

so x = -13 ± 12

so that x = -1 or x = 25

The number 25 was chosen deliberately so that, after adding 169 there would be a perfect square. But the method works equally well otherwise.

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That is all true, but not the original equation and certainly not the most efficient way to solve it. I never claimed "completing the squares" was not "a very powerful way of solving general quadratic equations". I simply stated, "I would be remiss if I did not mention that this is not the best method for solving this equation."