The decision and its conditions
Completing a square turns a quadratic expression into a squared binomial plus a constant. Half the linear coefficient before squaring it; adding a completion term to an equation requires balancing the other side. The discriminant tells how many distinct real roots exist, not their values. A product of roots and a turning-point input answer other questions and may be obtained without solving every root. For a downward-opening model, the turning point gives a maximum. Keep the domain of an application separate from the algebraic features of its formula.
Worked example
What constant c makes x² + 8x + c a perfect square trinomial? A square (x+h)² expands to x²+2hx+h². Here 2h=8, so h=4. The needed constant is h²=16.
A contrasting decision
How many distinct real solutions does x² + 4x + 3=0 have? The discriminant is 4²−4×3=4. A positive discriminant gives two distinct real roots. The requested count is 2, not a root value.
Diagnose a competing method
The first example’s tempting answer 4 follows this mistake: Report the shift inside the square rather than its square. Match the middle coefficient to twice a number before squaring that number.
Transfer before revealing
The two roots of x² − 11x + 28=0 are r and s. What is rs? Pause and identify the requested quantity. Solution: For a monic quadratic, the product of the roots equals the constant term. The factor constants are 4 and 7, whose product is 28. Thus rs=28.
Check the boundary of the method
For a downward-opening model, the turning point gives a maximum. Calculate the discriminant and classify its sign; solving both roots is unnecessary. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
What constant c makes x² + 8x + c a perfect square trinomial?
- A square (x+h)² expands to x²+2hx+h².
- Here 2h=8, so h=4.
- The needed constant is h²=16.
Compare the other answer choices
Divide the middle coefficient by four instead of squaring half of it.
Report the shift inside the square rather than its square.
Give the completion constant the wrong 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.
GED Mathematical Reasoning Assessment Guide for Educators