Square Roots and the Zero-Product Property

In this lesson you will: Solve x^2=q and (x-p)^2=q by taking square roots of both sides; Keep both signs: a positive number has two square roots; Use the zero-product property on a product that equals zero.

Two numbers square to the same thing

Solving x^2=49 means finding every number whose square is 49. There are two of them, 7 and -7, because squaring destroys the sign.

So the square-root step carries a +/-: x=+/-7. Writing only x=7 answers half the question, and on an equation item half an answer scores what no answer scores.

The same move works on a shifted square

(x-3)^2=25 is the same problem with x-3 playing the part of x. Take square roots: x-3=+/-5. Then finish the two cases separately, x=8 and x=-2.

The answer does not have to be a whole number. (x+4)^2=7 gives x+4=(7), so x=-4(7). Leave it in that exact form unless a context asks for a decimal, in which case x ~ -1.35 or x ~ -6.65.

Zero is the only number a product can be split at

If two numbers multiply to zero, at least one of them is zero. Nothing else can happen: a product of two nonzero numbers is nonzero.

That single fact, the zero-product property, turns one quadratic equation into two linear ones. From (x-2)(x+5)=0 you get x-2=0 or x+5=0, and the rest is Chapter 3 work.

Common mistake — Losing the negative solution

A student solves x^2=36 and writes x=6.

Every reason to do this is a good one. Your calculator returns 6. The square-root key has one output. Years of arithmetic have trained the response that sqrt(36) is 6, and that is correct — sqrt(36) really is 6.

The gap is between the square root and solving an equation. The symbol sqrt(36) names one number, the positive one. The equation x^2=36 asks a different question: which numbers square to 36? Two of them do.

The same slip on (x-1)^2=16 produces x=5 and hides x=-3 completely.

How to catch it: Count your answers. Any square-root step on a positive number should leave you holding two, and they should sit the same distance either side of p. Then test the negative one — it is the one you nearly threw away.