Q 6.104 - Intermediate Algebra

Question

The product of two consecutive odd integers is $483.$ Find the integers.

Step-by-Step Solution

Answer

The two consecutive odd integers are $21, 23 and -23, -21.$

Step 1. Given Information

There are two consecutive odd integers whose product is $483.$

We have to find the integers.

Step 2. Finding the integers

Let the first integer be $x$

and next consecutive odd integer be $x+2$

Therefore, the product of two consecutive odd integers is $\left(x\right)(x+2)=483.$

Step 3. Solve the equation

The equation we get is

$$\begin{gathered} \left(x\right)(x+2)=483 \\ {x}^2+2x-483=0 \\ {x}^2+23x-21x-483=0 \\ x(x+23)-21(x+23)=0 \\ (x-21)(x+23)=0 \end{gathered}$$

So,

 $$\begin{gathered} x-21=0 \\ x=21 \end{gathered}$$ and $$\begin{gathered} x+23=0 \\ x=-23 \end{gathered}$$

Step 4. Put the value of x in assumed integers to find the exact integers

If $x=21$

So, the first integer is $21$

And the next consecutive odd integer is

$21+2=23.$

If $x=-23$

So, the first integer is $-23$

And the next consecutive odd integer is

$-23+2=-21.$

Step 5. Verify the integers

As the product of two consecutive odd integers is $483.$

Let's verify the integers we get by multiplying them.

So, the two consecutive odd integers are $21,23$ and their product is $21\times 23=483.$

And another two consecutive odd integers are $-23,-21$ and their product is $(-23)\times (-21)=483.$

Thus, both pairs of consecutive odd integers are solutions.