Questions

1. What does it mean for $x$ to be a solution of an equation?
For example, the equation $(x-1)(x+2) = 0$ has solutions $x = 1$ or $-2$, and the equation $x – 2 = 3$ has the solution $x = 5$. What does this actually mean?

2. Joey attempted to solve the equation $x + 7 = 9$ and said that $x = 3$.
Without actually solving for $x$, how can you check to see if Joey’s answer is correct?

3. Explain why $9 \times 8 \times 7 \times 6 \times 0 = 0$, or why $999 \times 0 \times 777 \times 555 = 0$.
Then, solve the equation:

$$9 \cdot x \cdot 8 \cdot y \cdot (z – 2) = 0$$

using the same reasoning.

4. What does it mean to generalize in mathematics?
Why might it be useful to get a generalized solution?
Why might it not be useful to get a generalized solution?

Solutions

1.
For $x$ to be a solution of an equation, it means it’s a number that makes the equation true when you substitute it in.

Example: $(x-1)(x+2) = 0$
When $x = 1$:

$$(1-1)(1+2) = 0 \quad \Rightarrow \quad 0 \cdot 3 = 0$$

True, so $x = 1$ is a solution.

When $x = -2$:

$$(-2-1)(-2+2) = (-3)(0) = 0$$

True, so $x = -2$ is also a solution.

If $x = 3$:

$$(3-1)(3+2) = 2 \cdot 5 = 10$$

Not equal to 0, so $x = 3$ is NOT a solution.

2.
To check Joey’s answer without solving the equation yourself, plug it into the original equation.

For $x = 3$:

$$3 + 7 = 10$$

But the right side of the equation is $9$, and $10 \neq 9$.
This means Joey’s answer is incorrect.

3.
When you multiply anything by zero, the result is zero. This is why:

$$9 \cdot 8 \cdot 7 \cdot 6 \cdot 0 = 0$$

and

$$999 \cdot 0 \cdot 777 \cdot 555 = 0$$

For the equation:

$$9 \cdot x \cdot 8 \cdot y \cdot (z – 2) = 0$$

The product will be 0 if ANY factor is 0:

• $x = 0$

• $y = 0$

• $z – 2 = 0 \Rightarrow z = 2$

4.
To generalize means to solve a problem in a way that works for all possible examples at once, instead of just one set of numbers.

Example of specific case:
Solve $2x + 3 = 5$

$$2x = 2 \quad \Rightarrow \quad x = 1$$

Example of general case:
Solve $ax + b = c$ where $a \neq 0$:

$$ax = c – b \quad \Rightarrow \quad x = \frac{c – b}{a}$$

This works for any $a, b, c$.

Why useful: It creates one formula for every case (e.g., the quadratic formula).
Why not always useful: If you only care about one quick answer, generalizing may be unnecessary.