Quadratic Equations by Factoring and Square Roots

A quadratic equation is simply one with an $x^2$ in it, like $x^2 - 3x = 10$. ("Quadratic" comes from the Latin for "square.") The zero product property up top is the key: $ab = 0$ (two things multiplied give zero) means $a = 0$ or $b = 0$ (at least one of them was zero). The other formula, $x = \pm\sqrt k$, just says a number has two square roots, one positive and one negative. That's the whole lesson: two short facts.

Why factoring helps. In the last unit you learned to factor; here's the payoff. A linear equation could be undone step by step, but $x^2 - 3x - 10 = 0$ can't, because $x$ appears in two places. Factoring it into $(x - 5)(x + 2) = 0$ turns one hard equation into two easy ones, $x - 5 = 0$ and $x + 2 = 0$.

Square root property. For equations like $(x - 2)^2 = 9$, there's no need to multiply out. Something squared is 9, so that something is $3$ or $-3$: $x - 2 = \pm3$, giving $x = 5$ or $x = -1$.

A common mistake. It's tempting to factor whatever is there and solve, so $x(x - 3) = 4$ becomes "$x = 4$ or $x - 3 = 4$." That only works with zero on the right. Two numbers can multiply to 4 in endless ways (like $1 \times 4$ or $8 \times \frac12$), so knowing the product is 4 tells you nothing about either factor. Only zero forces a factor to be zero. Move everything to one side first: $x^2 - 3x - 4 = 0$.

Projectile landing times, the dimensions of a garden with a given area, and break-even points often come from factorable quadratics.