Equivalent forms reveal different features of the same function. Factored form reveals roots, vertex form reveals a minimum or maximum, and standard form reveals coefficients. Before expanding everything, decide which form answers the question most directly.
A reliable method
Identify the requested feature
A root is an input where the output is zero. A minimum is an output. Function evaluation asks for an output at a given input; it does not ask you to solve f(x) = 0.
Choose a useful form
For ax² + bx + c, factoring may reveal roots. Completing the square reveals the vertex. In A·b^(t/k), t/k counts the number of growth periods.
Record restrictions
A denominator cannot be zero. A real square root requires a nonnegative radicand. Squaring an equation can introduce solutions that do not satisfy the original equation.
Check in the original expression
Evaluate candidate roots before accepting them. For a minimum, explain why the squared term cannot be negative rather than relying only on a calculator image.
Worked examples
For f(x) = x² − 8x + 19, what is the minimum value?
- Complete the square: x² − 8x + 19 = (x − 4)² + 3.
- The square is nonnegative for every real x.
- It becomes zero at x = 4, so the smallest output is 3.
Answer: 3, attained at x = 4
Solve √(x + 6) = x for real x.
- The right side equals a principal square root, so x must be nonnegative.
- Square both sides: x + 6 = x². Rearrange and factor: (x − 3)(x + 2) = 0.
- Check 3: √9 = 3. Check −2: √4 = 2, which is not −2.
- Reject the extraneous root −2.
Answer: x = 3 only
Now use the method yourself.
Mark this guide as read, then use the method in topic practice before the checkpoint.