A reliable method
Read a vertex from a completed square; read roots from zero factors. For a monic quadratic, expand the factors to check both middle and constant terms.
The common trap
A product is zero only when at least one factor is zero. You cannot set individual factors equal to a nonzero right-hand side.
Make the connection
Physical lengths impose a positive-domain restriction even when an algebraic equation has two roots. Keep all candidates until the context selects one.
Worked example: choose the requested quantity
For real x, what is the minimum value of y = 2(x − 3)² + 4? The square is nonnegative for every real input. It becomes zero at the horizontal vertex coordinate, leaving only the vertical offset. The requested value is 4.
Contrast and diagnose an error
The roots of x² − 7x + 12 = 0 are r and s. What is rs? Factor as two linear factors with roots whose sum and product match the coefficients. For a monic quadratic, the root product is its constant term. The requested value is 12. A tempting answer is -12: Reverse the constant sign in the root product.
Transfer practice and check
For all x, x² + 7x + k = (x + 3)(x + 4). What is k? Try it before reading on. Check: 12. Expand the product and collect like powers. The constant term is the product of the two constants.
For real x, what is the minimum value of y = 2(x − 3)² + 4?
- The square is nonnegative for every real input.
- It becomes zero at the horizontal vertex coordinate, leaving only the vertical offset.
- The requested value is 4.
Compare the other answer choices
Report the horizontal vertex coordinate.
Multiply the offset by the leading coefficient.
Reverse the vertical shift sign.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content