Question

Find two consecutive integers whose product is 812.1. The smallest integer is 2. The largest integer is

Answer 1

We can answer this question as follows:

1. We have two consecutive integers, and we can express them as follows:

`x,x+1`

2. We need the product of them to be equal to 812:

`x(x+1)=812`

3. Now, we can expand the product as follows - we can apply the distributive property:

`\begin{gathered} x(x+1)=812 \\ x\cdot x+x\cdot1=812 \\ x^2+x=812 \end{gathered}`

4. We can subtract 812 from both sides of the equation, and we end up with a quadratic equation:

`\begin{gathered} x^2+x-812=812-812 \\ x^2+x-812=0 \end{gathered}`

5. We have to apply the quadratic formula to solve this equation as follows:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a},ax^2+bx+c=0`

And we have that:

`\begin{gathered} x^2+x-812=0 \\ a=1 \\ b=1 \\ c=-812 \end{gathered}`

6. Now, we can apply the quadratic formula:

`\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-1\pm\sqrt[]{(1)^2-4(1)(-812)_{}}}{2(1)} \\ x=\frac{-1\pm\sqrt[]{1^{}+3248_{}}}{2(1)} \end{gathered}`

Then we can continue to finally have two solutions:

`\begin{gathered} x=\frac{-1\pm\sqrt[]{3249_{}}}{2(1)} \\ x=(-1\pm57)/(2) \\ x_1=(-1+57)/(2)=(56)/(2)\Rightarrow x_1=28 \\ x_2=(-1-57)/(2)=(-58)/(2)\Rightarrow x_2=-29 \end{gathered}`

7. We can, now, check which of the two solutions are the correct one:

If we check with x = 28, we have:

`\begin{gathered} x(x+1)=812 \\ 28\cdot(28+1)=812 \\ 28\cdot29=812 \\ 812=812\Rightarrow It\text{ is True.} \end{gathered}`

If we check with x = -29, we have:

`\begin{gathered} -29\cdot(-29+1)=812 \\ -29\cdot(-28)=812 \\ 812=812\Rightarrow It\text{ is true.} \end{gathered}`

Therefore, we have two possible answers:

• 28 and 29

,

• -29 and -28

They are integers (one pair positive, and the other pair negative integers).

In summary, then we can say:

For 28 and 29:

• The smallest integer is 28

,

• The largest integer is 29

For -29 and -28

• The smallest integer is -29

,

• The largest integer is -28

[We need to remember that integers are all "positive and negative whole numbers: {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}.] {...