The sum of a number and its multiplicative inverse is negative two

Answer 1

Let's use x to represent the mystery number:

Now the multiplicative inverse, which we also refer to as the reciprocal will be (1/x), because,

x times (1/x) will equal 1.

Now we want to find out the value of x SUCH that the sum of x and its multiplicative inverse will equal -2.

The equation that expresses that is:

x+(1/x)=-2

Now we can rearrange this equation by multiplying both side by x

x²+1=-2x

and we can write this as a quadratic equation.

x² + 2x + 1 = 0

Now you can express this expression on the left as a product of two factors:

(x+1)(x+1) = 0

And there is only one root (x = -1) and you can see that fits the original equation

x+(1/x)=-2

Answer 2

Let x be the number we are trying to find.

The multiplicative inverse of that number is 1/x.

Now we just need to translate the word problem into an equation.

The sum of a number and its multiplicative inverse --> x + 1/x

is negative 2 ---> = -2

Putting the whole equation together: x + 1/x = -2

To solve this equation we first want to cancel out the x in the denominator, so we multiply both sides of the equation by x:

x (x + 1/x) = x (-2)

x**2 + 1 = -2x

then add 2x to both sides of the equation to set up the quadratic:

x**2 + 2x + 1 = 0

This is a perfect square trinomial which factors to:

(x+1)(x+1) = 0

In either case we have x+1 = 0

so x = -1.

To verify x + 1/x = =-1 + 1/-1 = -1 + -1 = 2