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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Quadratic Expressions and Equations: Unit Review

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2About 40 minutes
Learning goals
  • I can expand and simplify second-degree expressions and factor them in every way this unit teaches.
  • I can connect factors, zeros, vertex form and the graph of a quadratic relation.
  • I can solve quadratic equations by factoring, square roots, the quadratic formula and graphing, and use the discriminant.
  • I can find maximum and minimum values and solve real problems with quadratic models.

Words to know

quadratic expression
A second-degree polynomial such as ax² + bx + c.
factored form
y = a(x − r)(x − s); the zeros are r and s.
vertex form
y = a(x − h)² + k; the vertex is (h, k).
quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a.
discriminant
b² − 4ac; positive, zero or negative for two, one or no real roots.
optimization
Finding a maximum or minimum value in a problem.

Let's review!

In this unit you built the algebra that goes with the graphs of quadratic relations. You expanded second-degree expressions, factored them in every way the course asks for (common factors, grouping, simple and complex trinomials, differences of squares and perfect squares), and connected factors to zeros. You solved quadratic equations by factoring, square roots, the quadratic formula and graphing, rewrote relations in vertex form by completing the square, and used the discriminant to tell how many real roots an equation has. Finally you found maximum and minimum values and solved real problems about thrown objects, areas and revenue.

This review has a few questions from every lesson. Try each one on paper, then press Show answer to check.

How to use this review

Do two or three sections at a time rather than all at once. Write full solutions, not just answers, because most marks in Grade 10 math are for the steps. For each mistake, decide why it happened (sign error? forgot a common factor? wrong method?) and go back to that lesson's worked examples. Keep a calculator handy for the formula questions, and a graphing tool if you have one. Finish with the 10-question quiz.

Standard formy = ax² + bx + cshows the y-intercept cFactored formy = a(x − r)(x − s)shows the zeros r and sVertex formy = a(x − h)² + kshows the vertex (h, k)factorexpandcomplete the squareexpandsame parabola,three equations
The three forms of a quadratic relation and how to move between them. Expanding always works; factoring works when the expression factors; completing the square always works.

Expanding and Simplifying Second-Degree Expressions

1

Expand (3x − 4)².

2

Expand and simplify (x + 5y)(2x − y).

3

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

Factoring with Common Factors and by Grouping

1

Factor 9x³ − 12x² fully.

2

Factor 5(y − 2) − x(y − 2).

3

Factor xy + 3x + 2y + 6 by grouping.

Factoring Trinomials of the Form x² + bx + c

1

Factor x² − 4x − 32.

2

Factor x² + 14x + 48.

3

Factor 2x² − 20x + 42 fully.

Factoring Trinomials of the Form ax² + bx + c

1

Factor 3x² + 11x + 6.

2

Factor 4x² − 3x − 10.

3

Factor 6x² − 13x + 6.

Differences of Squares, Perfect Squares and Factoring Fully

1

Factor 49x² − 1.

2

Factor 25x² − 20x + 4.

3

Factor 3x³ − 27x fully.

4

Can x² + 16 be factored over the integers?

−101234567−6−4−20246xyABCD
Use this graph for the next set of questions. Points A, B, C and D are labelled on the parabola y = −x² + 6x − 5.

Factored Form: Connecting Factors and Zeros

1

For the graph above, write the equation in factored form and give the coordinates of A, B, C and D.

2

For y = 3(x − 2)(x + 4), find the zeros, the vertex and the y-intercept.

3

Find the equation of the parabola with zeros 1 and 5 that passes through (0, −10).

Solving Quadratic Equations by Factoring

1

Solve x² + x − 42 = 0.

2

Solve 6x² = 15x.

3

Solve 4x² + 4x − 3 = 0.

Completing the Square: From Standard Form to Vertex Form

1

Write y = x² − 10x + 18 in vertex form.

2

Write y = 3x² + 12x + 7 in vertex form.

3

Write y = −2x² + 8x + 1 in vertex form and state the maximum value.

Sketching y = ax² + bx + c: Choosing a Method

1

Sketch y = x² + 2x − 15: give the zeros, vertex and y-intercept.

2

Use partial factoring to find the vertex of y = x² − 4x + 9.

3

Which form shows the vertex directly?

The Quadratic Formula: Where It Comes From and How to Use It

1

Solve x² − 3x − 5 = 0 to two decimal places.

2

Solve 2x² + x − 2 = 0 to two decimal places.

3

In the development of the quadratic formula, which method is applied to ax² + bx + c = 0?

Solving Quadratic Equations: Square Roots, Graphing and Choosing a Method

1

Solve (x − 6)² = 49.

2

Solve 3(x + 2)² − 75 = 0.

3

Find the intersection points of y = x² and y = x + 6.

Real and Non-Real Roots: The Discriminant

1

How many real roots does 4x² + 4x + 1 = 0 have?

2

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

3

For what values of k does x² + kx + 4 = 0 have two real roots?

Zeros and Maximum or Minimum Values

1

Find the maximum value and the zeros of y = −x² + 8x − 7.

2

Two numbers have a sum of 16. What is their largest possible product?

3

Find the minimum value of y = 3x² − 18x + 5.

Just for fun

Why did the quadratic equation sign up for the review session?

It had a few unresolved roots it wanted to sort out before the test.

Solving Problems: Heights of Thrown Objects

1

For h = −5t² + 40t + 3, find the starting height and the maximum height.

2

When does the object in h = −5t² + 40t + 3 land? (Two decimal places.)

3

For h = −5t² + 40t + 3, during what time interval is the height above 63 m?

Solving Problems: Area, Revenue and Number Puzzles

1

A rectangle's length is 3 m more than its width, and its area is 70 m². Find its dimensions.

2

A store sells 300 toques at $10. Each $1 increase loses 10 sales. Find the price for maximum revenue and the revenue.

3

The product of two consecutive integers is 56. Find them.

Check your understanding

Quiz

1.

Expand (2x − 3)².

2.

Factor x² − 5x − 24.

3.

Factor 6x² + x − 2.

4.

Factor 16x² − 81.

5.

What are the zeros of y = −2(x − 3)(x + 5)?

6.

Which is the vertex form of y = x² + 8x + 10?

7.

Solve x² − 7x = 0.

8.

How many real roots does x² + 2x + 5 = 0 have?

9.

Solve x² − 4x − 1 = 0 to two decimal places.

10.

A ball's height is h = −5t² + 30t + 2. What is its maximum height?

For parents and teachers

How to use the review results:

  • Errors expanding, especially (x + a)² written as x² + a², or sign errors when subtracting a product: revisit Lesson 1 (Expanding and Simplifying).
  • Not taking out the greatest common factor, or grouping brackets that do not match: Lesson 2 (Common Factors and Grouping).
  • Wrong signs when factoring x² + bx + c: Lesson 3.
  • Trouble with ax² + bx + c (using c instead of ac, or dropping the 1 in a grouping): Lesson 4.
  • Trying to factor a sum of squares, or stopping before factoring fully: Lesson 5.
  • Sign errors between factors and zeros, or finding the vertex from factored form: Lesson 6.
  • Forgetting to make one side 0, or dividing by x and losing a root: Lesson 7.
  • Completing the square when a ≠ 1 (forgetting to multiply by a): Lesson 8.
  • Choosing a sketching method or using the step pattern: Lesson 9.
  • Sign errors with −b or b² in the quadratic formula: Lesson 10.
  • Forgetting ± with square roots, or reading y-values instead of x-values at intersections: Lesson 11.
  • Discriminant arithmetic or interpreting "no real roots": Lesson 12.
  • Mixing up where a max or min occurs (x) with its value (y): Lesson 13.
  • Height problems: solving h = 0 instead of h = a given height, or keeping negative times: Lesson 14.
  • Setting up area and revenue models (2x for borders, price versus number of increases): Lesson 15.

This unit, together with Quadratic Relations and Their Graphs, completes the quadratic strand of MPM2D. These skills are used constantly in Grade 11 functions courses (MCR3U or MCF3M), where quadratic relations are studied as functions, and in physics.