Use the Entire Quadratic Formula
For Ax²+Bx+C=0 with A nonzero, roots are [−B±√(B²−4AC)]/(2A).
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: Solve 2x²−10x+12=0 using the quadratic formula. Candidate answers: (A) x=2 or x=3 (B) x=4 or x=6 (C) x=5 only (D) x=−2 or x=−3.
- Independent check inputs: Solve 2x²−14x+24=0 using the quadratic formula. Candidate answers: (A) x=3 or x=4 (B) x=6 or x=8 (C) x=7 only (D) x=−3 or x=−4.
Learn the idea
Quadratic equations and graphs express the same underlying structure. A discriminant predicts real-root count, the quadratic formula calculates roots, and vertex form displays shifts, direction and stretching directly. For Ax²+Bx+C=0 with A nonzero, roots are [−B±√(B²−4AC)]/(2A). Identifying signed coefficients first prevents a mistaken numerator or discriminant.
Here is the method on the supplied teaching case. The discriminant is [-10]²−4(2)(12)=4. Thus x=[10±2]/4, giving 2 and 3. Keep the full numerator over 2A.
The entire numerator is divided by 2A. Preserve the signed B value, and calculate the discriminant before the square root; both plus and minus branches require checking.
- 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: Solve 2x²−10x+12=0 using the quadratic formula. Candidate answers: (A) x=2 or x=3 (B) x=4 or x=6 (C) x=5 only (D) x=−2 or x=−3. Solution: The discriminant is [-10]²−4(2)(12)=4. Thus x=[10±2]/4, giving 2 and 3. Keep the full numerator over 2A. The final answer is x=2 or x=3. The entire numerator is divided by 2A. Preserve the signed B value, and calculate the discriminant before the square root; both plus and minus branches require checking.
Check your reasoning
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Helpful tips
- For Ax²+Bx+C=0 with A nonzero, roots are [−B±√(B²−4AC)]/(2A).
- The entire numerator is divided by 2A.
See the answer and explanation
x=3 or x=4
The discriminant is [-14]²−4(2)(24)=4. Thus x=[14±2]/4, giving 3 and 4. Keep the full numerator over 2A.
Take it further
Changed-input task: Solve 2x²−38x+180=0 using the quadratic formula. Candidate answers: (A) x=18 or x=20 (B) x=19 only (C) x=9 or x=10 (D) x=−9 or x=−10. Show the mathematical reason, not only the selected answer. Expected result: x=9 or x=10. The discriminant is [-38]²−4(2)(180)=4. Thus x=[38±2]/4, giving 9 and 10. Keep the full numerator over 2A.
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