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II-01: Multistep algebra and quadratic equations

IntoChemistry | General Chemistry II Math Readiness

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Module II-01 · Recommended before: Before equilibrium ICE-table calculations

Multistep algebra and quadratic equations

Chemical equilibrium may lead to quadratic expressions. Use the quadratic formula, then check which solution is physically meaningful.

Learn the method

a x²+b x+c=0

  1. Write an equation equal to zero.
  2. Identify coefficients a, b and c.
  3. Apply x = (−b ± √(b²−4ac))/(2a).
  4. Check both algebraic roots.
  5. Reject roots producing negative concentrations or exceeding the initial amount.

Common mistake: A mathematically valid quadratic root may be chemically impossible.

Worked example

Problem: Solve x² − 5x + 6 = 0.

Step-by-step solution: (x − 2)(x − 3) = 0, so x = 2 or 3. Additional chemical constraints would determine the acceptable root.

Check the units and whether the answer is reasonable before continuing.

More applications of this mathematics

Try each extension before expanding its solution.

Physical root selection

If x²/(0.100−x)=0.050, solving x²+0.050x−0.00500=0 gives x=0.050 M or −0.100 M. Only 0.050 M is physically possible.

Simultaneous constraints

For A ⇌ B initially 0.100 M A, if x transforms to B, [A]=0.100−x and [B]=x, with 0≤x≤0.100.

Independent practice — five questions

Suggested mastery threshold: four out of five. Incorrect answers display a worked explanation after submission.

Choose one response for each question, then select “Check my answers.” No timer. Hints are optional. You can retry freely. Questions are formative, not proctored.

Question 1. Solve x² -12x + 35 = 0. What is the smaller root?
Hint for question 1

Two algebraic roots may exist; compare them.

Question 2. Solve x² -14x + 48 = 0. What is the smaller root?
Hint for question 2

Two algebraic roots may exist; compare them.

Question 3. Solve x² -16x + 63 = 0. What is the smaller root?
Hint for question 3

Two algebraic roots may exist; compare them.

Question 4. Solve x² -18x + 80 = 0. What is the smaller root?
Hint for question 4

Two algebraic roots may exist; compare them.

Question 5. Solve x² -20x + 99 = 0. What is the smaller root?
Hint for question 5

Two algebraic roots may exist; compare them.

Supporting reference

OpenStax Chemistry 2e / NIST — reference for this subject. This lesson’s wording, examples and questions are original.

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