Question

If \((a-3)^2 + |b-3| = 0\), what is the value of a - b?

Hint:

Whenever you see an equation where a sum of non-negative terms (like squares, absolute values, or even roots of non-negative numbers) equals zero, you can immediately set each term equal to zero. This is a powerful shortcut for solving such problems.

Answer 1

Step 1: Understanding the Concept:
The problem involves an equation with a squared term and an absolute value term. A key property of real numbers is that the square of any real number is non-negative (\(\ge 0\)), and the absolute value of any real number is also non-negative (\(\ge 0\)). The sum of two non-negative numbers can only be zero if both numbers are themselves zero.
Step 2: Key Formula or Approach:
The given equation is \((a-3)\^{}2+|b-3|=0\).
Let \(X = (a-3)\^{}2\) and \(Y = |b-3|\). The equation is \(X+Y=0\).
We know that \(X \ge 0\) and \(Y \ge 0\).
The only way for the sum \(X+Y\) to be 0 is if \(X=0\) and \(Y=0\).
This gives us a system of two separate equations to solve for \(a\) and \(b\).
Step 3: Detailed Explanation:
From the principle described above, we must have:
1) \((a-3)\^{}2 = 0\)
2) \(|b-3| = 0\)
Let's solve the first equation for \(a\):
\[ (a-3)\^{}2 = 0 \] Taking the square root of both sides:
\[ a-3 = 0 \] \[ a = 3 \] Now, let's solve the second equation for \(b\):
\[ |b-3| = 0 \] The absolute value of an expression is zero only if the expression itself is zero.
\[ b-3 = 0 \] \[ b = 3 \] We have found the values of \(a\) and \(b\). The question asks for the value of \(a-b\).
\[ a - b = 3 - 3 = 0 \] Step 4: Final Answer:
The value of \(a-b\) is 0.