For what values of a the equation a ^ 2-4 = (a-2) x has infinitely many roots.

1. Let us factor out the difference of squares using the corresponding formula for reduced multiplication:

a ^ 2 – 4 = (a – 2) x;
(a – 2) x = a ^ 2 – 4;
(a – 2) x = (a + 2) (a – 2).
2. For a = 2 we get the identity:

0 * x = 0,

therefore, the equation in this case has infinitely many roots.

3. For a ≠ 2, we obtain an equation with a single root:

(a – 2) x = (a + 2) (a – 2);
x = (a + 2) (a – 2) / (a – 2);
x = a + 2.
Answer. The equation has an infinite number of roots for the parameter value: a = 2.