These are last year’s equations in second place. I advise you to review in your course last year the exercises that you had solved and the method of resolution.

You can also do all exercises 1 to 7 on page 38 and 86 to 89 on page 46.

Click on the link to find the solutions.

How to solve an equation that seems more complicated, with denominators, large bars of fraction, “less” in front of these bars of fraction, etc.

Here are some other equations …. a little more complicated.

Here are the solutions:

So far, we are solving equations that admit a single solution. We can say that they are “determined” equations.

Special equations

1) Impossible equations

Either solve the equation: 0x = 4

Question : Is there a number that multiplies by 0 gives 4 as a product? None, of course! 0 being absorbent element for multiplication! This equation does not admit any solution. We will say that it is an impossible equation.

Here is a series of impossible equations.

0x = -3 0x = 78 0x = -56

2) Indeterminate equations

Either solve the equation: 0x = 0

Question : Is there a number that multiplies by 0 gives 0 as a product? Yes of course ! Any number multiplied by 0 gives 0 as product. Any number is therefore solution of this equation. Here is an equation that admits an infinity of solutions. It’s an indeterminate equation .

In summary then

An equation has one solution: determined equation

an equation admits an infinity of solutions: indeterminate equation

An equation admits no solution: impossible equation

2. The zero-product equations

Reminder : 0 is an absorbing element for multiplication. This means that if a factor of a product is zero, the product is necessarily zero.

Example : 3.5×2,7x0x3,6 = 0 because one of the factors of this product is zero.

Mathematically, whatever the numbers a and b,

If a = 0 or b = 0 then ab = 0

Reciprocally , we will admit that if a product is zero then at least one of its factors is necessarily zero.

Mathematically, whatever the numbers a and b,

if ab = 0, then a = 0 or b = 0

Example :

Here is a zero product: (x + 2) (x + 3) = 0. This product has two factors. Since the product is zero, it is because one of the two factors of the product is necessarily nil. Let the first factor (x + 2) = 0, or the second factor (x + 3) = 0. The product (x + 2) (x + 3) is therefore zero if x = -2 or x = -3.

The numbers found, -2 and -3 are solutions of the equation (x + 2) (x + 3) = 0.

An equation of this type is called a “zero product equation”.

Let’s generalize :

An equation of the type (ax + b) (bx + c) = 0 is called a zero product equation.

The solutions of such an equation are the solutions of each of the equations ax + b = 0 and cx + d = 0

(See the little summary at the bottom of page 32.)

How to reduce an equation to a zero product equation?