দ্বিঘাত সমীকরণের সমাধান: গুণফল শূন্য হবার নীতি

The zero product property tells us that if the product of any number of expressions is zero, at least one of them must be zero: $$a\cdot b\cdot c=0 \text{ means that } a=0 \text{ or } b=0 \text{ or }c=0$$ This is why we solve quadratic equations by first setting them equal to zero!

$$ax^2+bx+c=0$$

Now, we can factor this polynomial into terms. These terms will have a product of zero, so we'll find the roots of our equation by the zero product property.

Let's look at an example.

$$x^2+5x=-4$$

First, we set everything equal to zero. Here, we add $4$ to both sides.

$$x^2+5x+4=0$$

Then, we factor the left hand side:

$$(x+4)(x+1)=0$$

Since the expressions $x+4$ and $x+1$ multiply together to give $0$, we know that at least one of them equals zero. This lets us find solutions to our original equation:

$$\begin{alignat*}{2} x+4&=0 \rightarrow x&&=-4\\ x+1&=0 \rightarrow x&&=-1 \end{alignat*}$$

So our solutions for $x$ (or roots) are $-4$ and $-1$. We can plug these into the original equation to check. $$x^2+5x=-4$$ $$(-4)^2+5(-4)=16-20=-4 \quad \checkmark$$ $$(-1)^2+5(-1)=1-5=-4 \quad \checkmark$$

If you are unable to find two terms, it's possible you have a quadratic with no or infinitely many solutions!

The zero-product property is what allows us to find the zeroes of a polynomial by factoring it. Consider the expression $$a\cdot b = 0.$$ Intuitively it makes sense that in order for the product $a\cdot b$ to equal $0$, at least one of $a$ or $b$ must be $0$. So, if we consider a factored polynomial like $$x(x+1)(x-1)^2 = 0,$$ we can find the roots (also conveniently known as zeros) of the polynomial by setting each individual root equal to $0$.

• The first factor gives no trouble — $x = 0$.
• The second tells us that $x+1 = 0$, so $x = -1$.
• The last one gives us $x-1 = 0$, so $x = 1$. This root has multiplicity $2$, stemming from the fact that $x-1$ is squared in the factorization.

Now for some practice:

Good job! The next step is to actually factor quadratics that aren't already in factored form, so that you can then find the roots this way.

Zero Product Property

The Zero Product Property states that if $ab=0,$ then either $a=0, b=0,$ or both $a$ and $b$ equal $0.$

This property will be very helpful when we want to solve factored quadratic equations!

(Video) Zero Property

by GVSUmath

This video by GVSUmath explains the zero product property.

$\large{\text{Summary}}$

The zero product property states:

$A \cdot B = 0$ if and only if at least one of $A$ or $B$ is equal to $0$. .

We use this property when we solve quadratic equations. You may have noticed that we always manipulate quadratic equations to $ax^2+bx+c=0$. This is because factoring the equation gives us two expressions that multiply to zero. We can set each factor equal to zero and solve for $x$.

Let's assume we've factored a quadratic into $(x-1)(x+3)$. We'll solve it by setting each part equal to $0$. $$\begin{align} (x-1)(x+3)&=0\\ \\ x-1&=0\\ x&=1\\ \\ x+3&=0\\ x&=-3 \end{align}$$

Let's check our answers by plugging them into the equation $$\begin{align}(x-1)(x+3)&=0\\ \\ (1-1)(1+3)&=0\\ (0)(4)&=0\\ 0&-0\\ \\ (-3-1)(-3+3)&=0\\ (-4)(0)&=0\\ 0&=0 \end{align}$$

The equation holds true, so our solutions are $x=1, -3$.

Try one on your own:

We can solve each factor by setting it equal to $0$. $$\begin{align}(2x-9)(3x-9)&=0\\ \\ 2x-9&=0\\ 2x&=9\\ x&=\frac92\\ \\ 3x-9&=0\\ 3x&=9\\ x&=3\end{align}$$

The solutions are $x=\frac92, 3$.