Question about factoring a quadratic equation

Question

I was wondering if someone could clear this question I had.

$$x^2+4x-21=0$$ Factoring it out: $$(x+7)(x-3)$$ How does that mean that $x=-7$ or $x=3$?

How is it that we can just say "Alright, let's just forget one part of the equation $(x-3)$, and solve for $x$, or forget $x+7$ and solve for $x$," and it's going to hold true?

Thanks!!

Answer 1

Firstly, you probably meant $ x^2 - 4x - 21 = 0 $ as the equation $ x^2 + 4x - 21 = 0 $ does not have the factors you mentioned.

Now you have: $ (x+3)(x-7) = 0 $ . When the product of two numbers is zero, then either or both of them is zero, which means that,

$ (x+3) = 0 $, which implies that $ x = -3 $ , or $ (x-7) = 0 $ , i.e, $ x = 7 $ .

Answer 2

solving $$x^2+4x-21=0$$ we get $$x_{1,2}=-2\pm\sqrt{25}$$ and form here we get..... we get $x_1=-7,x_2=3$ so we have $$(x+7)(x-3)=x^2+4x-21$$

Answer 3

Prove that if $ab=0$, then $a=0$ or $b=0$.

Answer 4

The technical term is the Zero Factor Theorem. It states that if two (or more) numbers multiply to make zero, then at least one of them had to already be zero - in other words, if I multiply together a bunch of things that aren't zero, I can't get zero out.

In this case, factoring tells us that $x^2 + 4x - 21$ is the same as $(x - 7)(x + 3)$. So if $x^2 + 4x - 21 = 0$, then we know that $(x - 7)(x + 3) = 0$. By the Zero Factor Theorem, that means we have one of two situations: either $x - 7 = 0$ or $x + 3 = 0$. If $x - 7 = 0$, then $x = 7$. If $x + 3 = 0$, then $x = -3$.

Answer 5

We have to be careful, in which domain we solve such an equation. For example, in $R=\mathbb{Z}/4$ we have $2\cdot 2=0$, but none of the factors is zero. In a field $K$ however we have that $ab=0$ always implies that $a=0$ or $b=0$. Hence $$ (x+7)(x-3)=0 $$ implies then that $x+7=0$ or $x-3=0$.

Answer 6

$x^2 +4x -21=0.$

$x^2 +4x -21 = (x+7)(x-3) =0.$

The product $(x+7)(x-3) = 0$

$\iff $

one of the factors

$(x+7)$ or $(x-3)$ is 0.

1) $x+7= 0$ gives $x=-7;$

2) $x-3 = 0$ gives $x=3.$

Note:

For real $a,b:$

$ab = 0$ implies $a=0$ or $b=0$.

('Or' means : One factor or the other or both)

Helps?

Answer 7

This is how I learned it. Factoring an equation gives you the answers, as to where on the curve is $f(x)=0$. So when you factorize an equation, $x^2+4x−21=0$, you want to find which value of $x$ satisfies $f(x)=0$. In this case your equation was $x^2+4x−21=(x+7)(x-3)$. Then you consider each bracket separately, $x+7=0$ and $x-3=0$, $x_1=-7, x_2=3$. So your answers are $x_1=-7, x_2=3$. You can consider each bracket individually because if one of the brackets ends up as a $0$ then $f(x)$ will be $0$.