Which equations have both x = 5 and x = -5 as solutions?

To determine which equations have both x = 5 and x = -5 as solutions, we need to consider the characteristics of the equations that yield these values.

A polynomial equation can have multiple roots, and if we want both x = 5 and x = -5 to be solutions, we can formulate an equation using these roots.

For instance, we can create a quadratic equation where these are the roots. The general form of a quadratic equation with roots r1 and r2 can be expressed as:

y = (x - r1)(x - r2)

Substituting our roots:

y = (x - 5)(x + 5)

Now, if we expand this, we get:

y = x^2 - 25

Thus, one example of an equation where both x = 5 and x = -5 are solutions is:

y = x^2 - 25 = 0

When set to zero, we find:

x^2 - 25 = 0

This equation can be factored to:

(x - 5)(x + 5) = 0

Thus, the solutions are:

• x = 5
• x = -5

In conclusion, equations such as y = x^2 – 25 are examples where both x = 5 and x = -5 act as solutions. This principle also extends to higher-degree polynomials where both values are roots, provided the polynomial is constructed accurately to include them.