Supply the Term that Completes a Square
A square (x+a)² expands to x²+2ax+a².
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: What constant makes x²+4x+___ a perfect-square trinomial? Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
- Independent check inputs: What constant makes x²+6x+___ a perfect-square trinomial? Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9.
Learn the idea
Quadratic equations can be solved by revealing factors, making a square, or undoing a square. Each method exposes structure, and all valid roots must be retained and checked in the original equation. A square (x+a)² expands to x²+2ax+a². For a monic quadratic, halving the linear coefficient identifies a, and squaring that value supplies the completing constant.
Here is the method on the supplied teaching case. Half the x coefficient is 2; square it to get 4. The trinomial becomes (x+2)². Adding it to an equation requires adding the same amount on both sides.
When completing an equation’s square, add the constant to both sides. The needed constant is the square of half the linear coefficient, not half its square or the coefficient itself.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: What constant makes x²+4x+___ a perfect-square trinomial? Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: Half the x coefficient is 2; square it to get 4. The trinomial becomes (x+2)². Adding it to an equation requires adding the same amount on both sides. The final answer is 4. When completing an equation’s square, add the constant to both sides. The needed constant is the square of half the linear coefficient, not half its square or the coefficient itself.
Check your reasoning
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Helpful tips
- A square (x+a)² expands to x²+2ax+a².
- When completing an equation’s square, add the constant to both sides.
See the answer and explanation
9
Half the x coefficient is 3; square it to get 9. The trinomial becomes (x+3)². Adding it to an equation requires adding the same amount on both sides.
Take it further
Changed-input task: What constant makes x²+18x+___ a perfect-square trinomial? Candidate answers: (A) 162 (B) 80 (C) 81 (D) 82. Show the mathematical reason, not only the selected answer. Expected result: 81. Half the x coefficient is 9; square it to get 81. The trinomial becomes (x+9)². Adding it to an equation requires adding the same amount on both sides.
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More practice: try new cases
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