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A. Quadratic Functions · Quadratic Expressions and Equations

Quadratic Expressions and Equations: Unit Review

A1.1 A1.2 A1.3 A1.4 A1.5 A1.6 A1.7 A1.8About 40 minutes
Learning goals
  • I can expand, simplify and factor quadratic expressions, and represent situations with them.
  • I can solve quadratic equations by factoring, square roots, the quadratic formula and graphing, and choose a good strategy.
  • I can connect roots to x-intercepts and use the discriminant.
  • I can solve and interpret real problems.

Words to know

quadratic expression
An expression ax² + bx + c with a ≠ 0.
factor fully
Write as a product in which no factor can be factored further.
root / zero / x-intercept
The same numbers seen as an equation's solutions, a relation's zeros and a graph's x-intercepts.
quadratic formula
x = (−b ± √(b² − 4ac)) / (2a).
discriminant
b² − 4ac: positive, two real roots; zero, one; negative, none.

Let's review!

In this unit you worked with quadratic expressions and equations. You read quadratic relations from tables and graphs, represented situations with expressions, expanded and factored them (including ax² + bx + c, differences of squares and perfect squares), and solved equations by factoring, square roots, completing the square, the quadratic formula and graphing. You connected roots to x-intercepts, used the discriminant to count real roots, compared strategies and solved real problems.

This review has a short set of questions for every lesson. Try each one before pressing Show answer.

How to use this review

Work through one or two sections at a time. Show every step, and check each answer by substituting (for roots) or expanding (for factoring). If you miss a question, note why (sign error? forgot to rearrange? wrong method?) and reread that lesson's worked examples. Finish with the 10-question quiz. Keep a calculator handy for the formula questions and round final answers to two decimal places.

Expressionsexpand: (a ± b)², (a + b)(a − b)represent situationsfactor: GCF, x² + bx + c,ax² + bx + c, special productsSolvingrearrange to ax² + bx + c = 0factor + zero product propertysquare roots, completing the squarex = (−b ± √(b² − 4ac)) / (2a)Graphsroots = zeros = x-interceptsfactors give the interceptsD > 0: two; D = 0: one; D < 0: noneread tables and graphsProblemsdefine x, draw, write the equationchoose: factor, formula or graphreject impossible rootscheck and answer in a sentence
The four big ideas of the unit and the key tools in each.

Quadratic Relations in Tables and Graphs: Posing and Solving Problems

1

A table has x: 0, 1, 2, 3, 4 and y: 2, 7, 10, 11, 10. Is the relation quadratic? Where is the maximum?

2

A ball's height graph crosses h = 8 m at t = 0.5 s and t = 2.5 s. When is it highest?

3

Pose one question that a graph of revenue against price can answer.

Expanding and Simplifying Quadratic Expressions

1

Expand and simplify (2x + 5)(x − 4).

2

Expand and simplify 2x(x − 1) − (x − 3)².

3

Expand (4x − 3)(4x + 3).

Representing Situations with Quadratic Expressions

1

A 30 cm by 20 cm photo has a frame of width x. Write the total area in standard form.

2

Tickets cost $10 and 400 sell. Each $1 increase loses 25 sales. Write the revenue after x increases.

3

A rectangle's length is 3 more than twice its width w. Write its area.

Factoring: Common Factors and Trinomials x² + bx + c

1

Factor x² − 11x + 28.

2

Factor fully: −3x² + 12x + 36.

3

Factor 5x(x + 2) − 3(x + 2).

Factoring Trinomials ax² + bx + c

1

Factor 6x² + x − 12.

2

Factor 8x² − 14x + 3.

3

Factor 4x² + 4x − 15.

Differences of Squares, Perfect Squares and Choosing a Strategy

1

Factor fully: 18x² − 50.

2

Factor 25x² − 30x + 9.

3

Does x² + 49 factor? Explain.

Solving Quadratic Equations by Factoring

1

Solve x² − 4x − 21 = 0.

2

Solve 3x² − 5x = 2.

3

Solve x² = 5x.

−4−202468−27−18−909xy(−3, 0)(7, 0)y = x² − 4x − 21= (x − 7)(x + 3)
One parabola, three views: the equation x² − 4x − 21 = 0 has roots 7 and −3; the factors (x − 7)(x + 3) give them; and the graph crosses the x-axis there, with the axis of symmetry halfway between, at x = 2.
Just for fun

Quadratic equations have been solved for thousands of years: clay tablets from ancient Babylon show methods that are close relatives of completing the square. Your calculator is just the newest member of a very old club.

Factors and x-Intercepts: Connecting Equations and Graphs

1

Find the x-intercepts of y = 2x² + 7x − 4.

2

Write the equation of a parabola with zeros −3 and 5 that passes through (1, −32).

3

y = (x − 6)² has how many x-intercepts?

Completing the Square and Developing the Quadratic Formula

1

Solve (x − 3)² = 12 exactly.

2

Complete the square and solve: x² − 6x − 3 = 0.

3

In developing the quadratic formula, what is done to ax² + bx + c = 0?

Using the Quadratic Formula

1

Solve 2x² + 3x − 4 = 0.

2

Solve 2x² = 5x + 10.

3

A ball's height is h = −4.9t² + 20t + 2. When does it land?

Real Roots, x-Intercepts and the Discriminant

1

How many real roots does 3x² − 4x + 2 = 0 have?

2

For what k does x² − kx + 4 = 0 have exactly one root?

3

A height equation h = 25 has D = −3. What does that mean?

Choosing a Strategy: Graphing, Factoring or the Quadratic Formula

1

Which method would you use for (x + 2)² = 7? For x² − 3x − 10 = 0? For 1.5x² − 2.2x − 4 = 0?

2

Give one advantage of the quadratic formula and one of factoring.

3

How can a graph help even when you solve algebraically?

Solving Problems with Quadratic Equations

1

A 12 m by 10 m rink is surrounded by a walkway of uniform width; together they cover 168 m². How wide is the walkway?

2

One leg of a right triangle is 2 cm longer than the other; the hypotenuse is 10 cm. Find the legs.

3

Two consecutive positive integers have a product of 110. Find them.

Check your understanding

Quiz

1.

For equally spaced x-values, which tells you a relation is quadratic?

2.

Expand (x − 6)².

3.

A 20 cm by 12 cm card has a border of width x. The total area is:

4.

Factor fully: 3x² − 3x − 36.

5.

Factor 9x² − 16.

6.

Solve 2x² + 7x − 4 = 0.

7.

The x-intercepts of y = (x − 2)(3x + 1) are:

8.

Which is the quadratic formula?

9.

2x² − 3x + 5 = 0 has how many real roots?

10.

Which strategy is usually best for 0.6x² + 1.3x − 2.1 = 0?

For parents and teachers

Use the review to decide what to revisit:

  • Trouble reading graphs or tables, or with second differences → Quadratic Relations in Tables and Graphs.
  • Expansion errors, especially (a + b)² or subtracting a product → Expanding and Simplifying Quadratic Expressions.
  • Trouble writing expressions for frames, borders or revenue → Representing Situations with Quadratic Expressions.
  • Missing common factors or sign errors in x² + bx + c → Factoring: Common Factors and Trinomials x² + bx + c.
  • Errors with ax² + bx + c → Factoring Trinomials ax² + bx + c.
  • Missing difference-of-squares or perfect-square patterns, or not factoring fully → Differences of Squares, Perfect Squares and Choosing a Strategy.
  • Forgetting to rearrange to 0, dividing by x, or not verifying → Solving Quadratic Equations by Factoring.
  • Sign errors between factors and intercepts → Factors and x-Intercepts.
  • Losing the ± or making errors completing the square → Completing the Square and Developing the Quadratic Formula.
  • Formula arithmetic errors → Using the Quadratic Formula.
  • Miscounting roots or misreading "no real roots" → Real Roots, x-Intercepts and the Discriminant.
  • Always using one method → Choosing a Strategy.
  • Trouble setting up word problems or interpreting answers → Solving Problems with Quadratic Equations.

The companion unit, Quadratic Functions, builds directly on this one: function notation, domain and range, vertex form and transformations, and completing the square to find maximums and minimums. In Grade 12, Mathematics for College Technology (MCT4C) uses these skills again.