Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55

CAL2.32 Completing the Square Before Trig Substitution Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.32 Completing the Square Before Trig Substitution. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.32 Completing the Square Before Trig Substitution - Choose the correct completed-square form and substitution after shifting a quadratic expression. Standards: AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.32 Completing the Square Before Trig Substitution skill assessments:

Problem 1Difficulty: Mastery

Which setup for sqrt(9x² - 36x + 136) both completes the square and predicts the simplified radical after trig substitution?

A9x² - 36x + 136 = (3x - 6)² + 100; set 3x - 6 = 10 sin(theta); sqrt(9x² - 36x + 136) becomes 10 sec(theta)

B9x² - 36x + 136 = (3x + 6)² + 100; set 3x + 6 = 10 tan(theta); sqrt(9x² - 36x + 136) becomes 10 sec(theta)

C9x² - 36x + 136 = (3x - 6)² + 100; set 3x - 6 = 10 tan(theta); sqrt(9x² - 36x + 136) becomes 10 cos(theta)

D9x² - 36x + 136 = (3x - 6)² + 100; set 3x - 6 = 10 tan(theta); sqrt(9x² - 36x + 136) becomes 10 sec(theta)

Solution & Explanation:Since 9x² - 36x + 136 = (3x - 6)² + 100, the expression has the u² + a² form. Using set 3x - 6 = 10 tan(theta) makes sqrt(9x² - 36x + 136) simplify to 10 sec(theta) on the usual substitution interval. The simplification follows only after the shift is completed correctly.

Problem 2Difficulty: Mastery

Which choice correctly matches 25x² - 50x - 119 to a trig substitution and the radical factor it produces?

A25x² - 50x - 119 = (5x - 5)² - 144; set 5x - 5 = 12 sin(theta); sqrt(25x² - 50x - 119) becomes 12 tan(theta)

B25x² - 50x - 119 = (5x + 5)² - 144; set 5x + 5 = 12 sec(theta); sqrt(25x² - 50x - 119) becomes 12 tan(theta)

C25x² - 50x - 119 = (5x - 5)² - 144; set 5x - 5 = 12 sec(theta); sqrt(25x² - 50x - 119) becomes 12 tan(theta)

D25x² - 50x - 119 = (5x - 5)² - 144; set 5x - 5 = 12 sec(theta); sqrt(25x² - 50x - 119) becomes 12 sec(theta)

Solution & Explanation:Since 25x² - 50x - 119 = (5x - 5)² - 144, the expression has the u² - a² form. Using set 5x - 5 = 12 sec(theta) makes sqrt(25x² - 50x - 119) simplify to 12 tan(theta) on the usual substitution interval. The simplification follows only after the shift is completed correctly.

Problem 3Difficulty: Mastery

Complete the square before trig substitution. Which option correctly states the resulting simplified radical for sqrt(-16x² - 64x + 17)?

A-16x² - 64x + 17 = 81 - (4x + 8)²; set 4x + 8 = 9 tan(theta); sqrt(-16x² - 64x + 17) becomes 9 cos(theta)

B-16x² - 64x + 17 = 81 - (4x + 8)²; set 4x + 8 = 9 sin(theta); sqrt(-16x² - 64x + 17) becomes 9 cos(theta)

C-16x² - 64x + 17 = 81 - (4x - 8)²; set 4x - 8 = 9 sin(theta); sqrt(-16x² - 64x + 17) becomes 9 cos(theta)

D-16x² - 64x + 17 = 81 - (4x + 8)²; set 4x + 8 = 9 sin(theta); sqrt(-16x² - 64x + 17) becomes 9 sec(theta)

Solution & Explanation:Since -16x² - 64x + 17 = 81 - (4x + 8)², the expression has the a² - u² form. Using set 4x + 8 = 9 sin(theta) makes sqrt(-16x² - 64x + 17) simplify to 9 cos(theta) on the usual substitution interval. The simplification follows only after the shift is completed correctly.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55 (Choose the correct completed-square form and substitution after shifting a quadratic expression. Standards: AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55).