Question

How many solutions does the equation `x^2 - 16x + 64 = 0` have?

Answer 1

Conventionally, we say that the eqn. has `2` identical roots : `8&8`.

Explanation:

`x^2-16x+64+0 rArr (x-8)^2=0`

`rArr x=8, &, x=8`

Conventionally, we say that the eqn. has `2` identical roots : `x=8,8`.

Answer 2

For the quadratic equation `ax^2+bx+c=0`, the discriminant `Delta=b^2-4ac` tells us about the nature of the equation's roots:

• If `Delta>0`, the equation has two real roots.
• If `Delta=0`, the equation has a single double root.
• If `Delta<0`, the equation has no real solutions.

So, for `x^2-16x+64=0`, we see that `a=1`, `b=-16`, and `c=64`. Thus:

`Delta=b^2-4ac=(-16)^2-4(1)(64)=256-256=0`

Since `Delta=0`, the equation has one single solution.