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why is the quadratic formula called the quadratic formula when quad means four and x is only to the power of two and there are only three terms and you can use it more than four times.

this is very concerning.

please help, I cannot figure this out even when googling

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  • $\begingroup$ @H.sapiensrex I am pretty sure the verb "quadrare" and the number "quattuor" are themselves etymologically related. $\endgroup$
    – Ian
    Commented May 21 at 22:53
  • $\begingroup$ @Arthur im just guessing, but perhaps it has to do with how the formula is derived by completing the square? $\endgroup$
    – peek-a-boo
    Commented May 21 at 22:56

3 Answers 3

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It is called that because quadratic comes from the Latin word for square, and x is getting squared.

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    $\begingroup$ It derives from the Latin noun for a square, not the verb to square. $\endgroup$
    – Numeral
    Commented May 21 at 23:16
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A square has four sides, but its area is the side length raised to the second power. This is why raising to the second power ("squaring") is intimately linked to the Latin word for 4. And I believe we see this discrepancy most commonly in the adjective "quadratic", which linguistically seems to refer to a foursome of some kind, but mathematically means we have an exponent of 2.

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Historically, quadratic equations were viewed as equalities about areas of squares and rectangles. They were even treated differently depending on what we now call the signs of the coefficients. For example, $x^2=bx+c$ for $b,c>0$ was analyzed separately from $x^2+bx=c$ for $b,c>0$. That central role of squares and rectangles is the etymology of the word "quadratic"; indeed "quadrum" means "square" and "quadratum" means "having been made square" in Latin. These are both more distantly related to the word "quattuor" meaning "four".

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  • $\begingroup$ This is an excellent answer but while the study of quadrilaterals is important in the history of quadratics, the actual etymology derives from quadrate, the latin for square. $\endgroup$
    – Numeral
    Commented May 21 at 22:58
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    $\begingroup$ @Numeral Good catch. It really doesn't actually have anything to do with non-rectangular quadrilaterals, so I changed that. $\endgroup$
    – Ian
    Commented May 21 at 23:01