A. Characteristics of Functions · Functions and Equivalent Expressions

Are They Equivalent? Testing Algebraic Expressions

A3.4 A3.1 A3.2 A3.3About 70 minutes

Learning goals

I can explain what it means for two expressions to be equivalent.
I can prove that expressions are equivalent by simplifying them.
I can show that expressions are not equivalent with a counterexample.
I can account for restrictions when comparing rational expressions.
I can use √(ab) = √a × √b to show that two radical expressions are equivalent.

Words to know

equivalent expressions
Expressions with the same value for every value of the variable for which both are defined.

counterexample
A single example that shows a statement is false.

identity
An equation that is true for every allowed value of the variable, such as (x + 1)² = x² + 2x + 1.

mixed radical
A number times a radical, such as 6√2; equivalent to an entire radical such as √72.

hole
A missing point on a graph, at a value where an expression is undefined but a simplified form is not.

Same value, different look

"Twenty-five cents", "a quarter" and "$0.25" all name the same amount. Algebraic expressions can do the same thing: (x + 1)(x + 3), x² + 4x + 3 and (x + 2)² − 1 look different but give the same value for every x. They are equivalent.

In Grade 9 you compared expressions to see if they were equivalent. Now, with polynomials, radicals and rational expressions, you have more tools, and some new traps.

Definition

Two expressions are equivalent if they have the same value for every value of the variable for which both are defined.

There are three ways to investigate:

Simplify both expressions (expand, collect like terms, factor, reduce). If they become identical, they are equivalent. This is a proof.
Substitute values. If even one value gives different results, they are not equivalent. This is called a counterexample. But if a few values happen to agree, that is not a proof.
Graph both (with technology or by hand). Identical graphs are strong evidence; graphs that differ anywhere show they are not equivalent. Watch for holes where a rational expression is undefined.

Three ways to test equivalence. Simplifying is the only one that proves two expressions are equivalent.

Worked example

Example 1: Simplifying both sides

Are 8x − 2x(4x − 1) − 6 and −2(4x² − 5x + 3) equivalent?

Simplify the first: 8x − 8x² + 2x − 6 = −8x² + 10x − 6.
Simplify the second: −8x² + 10x − 6.
The simplified forms are identical, so the expressions are equivalent.
A quick extra check with x = 1: first 8 − 2(3) − 6 = −4; second −2(4 − 5 + 3) = −4 ✓.

Yes, they are equivalent.

Worked example

Example 2: One counterexample is enough

Are (x + 3)² and x² + 9 equivalent?

Substitute x = 1: (1 + 3)² = 16 but 1² + 9 = 10.
16 ≠ 10, so they are not equivalent.
Simplifying shows why: (x + 3)² = x² + 6x + 9. The middle term 6x is missing from x² + 9, so the two agree only when 6x = 0, that is at x = 0.

Not equivalent: x = 1 is a counterexample.

The graphs of (x + 3)² and x² + 9 cross only at (0, 9). Testing x = 0 alone would have fooled you.

Worked example

Example 3: Why agreeing values are not a proof

Lin tests x² + 2 and 3x at x = 1 and x = 2 and gets the same answers. She concludes they are equivalent. Is she right?

x = 1: 1 + 2 = 3 and 3(1) = 3. x = 2: 4 + 2 = 6 and 3(2) = 6. Both agree.
Try x = 0: 0 + 2 = 2 but 3(0) = 0. Not equivalent.
The two expressions are equal only when x² + 2 = 3x, that is x² − 3x + 2 = 0, which is (x − 1)(x − 2) = 0. Lin happened to test exactly the two values where they agree!

No. Agreeing at a few values does not prove equivalence; x = 0 is a counterexample.

Choosing test values

Avoid 0 and 1 as your only test values; they make many different expressions look alike. Try a negative number and a fraction too, such as x = −2 or x = 1/2. And never test a value that is a restriction.

Worked example

Example 4: Rational expressions and restrictions

Are (2x² − 2)/(x² + x) and 2(x − 1)/x equivalent?

Simplify the first: 2(x − 1)(x + 1)/[x(x + 1)] = 2(x − 1)/x, with restrictions x ≠ 0, −1.
The second expression has only the restriction x ≠ 0.
At x = −1, the second expression gives 2(−2)/(−1) = 4, but the first is undefined (0/0).
So they are equivalent for all x except x = 0 and x = −1. We write: (2x² − 2)/(x² + x) = 2(x − 1)/x, x ≠ 0, −1. Their graphs are identical except for a hole in the first at (−1, 4).

Equivalent with the restrictions x ≠ 0, −1.

Radicals: equivalent forms of numbers

Radicals have equivalent forms too. The key property is

√(ab) = √a × √b, for a ≥ 0 and b ≥ 0.

You can check it: √(4 × 9) = √36 = 6 and √4 × √9 = 2 × 3 = 6. A calculator agrees for any pair, such as √2 × √3 ≈ 2.449 ≈ √6. It does not work for addition: √9 + √16 = 7 but √(9 + 16) = 5.

The property lets you rewrite an entire radical such as √72 as an equivalent mixed radical in simplest form, by taking out the largest perfect square factor. Only like radicals (same radicand) can be added: 2√3 + 5√3 = 7√3, just like 2x + 5x = 7x. Products are expanded like polynomials, using √a × √a = a.

This is a short preview. The Quadratic Functions unit has a full lesson, Exact Answers: Simplifying and Operating with Radicals, where you will practise simplifying, adding, subtracting and multiplying radical expressions such as (2 + √6)(3 − √12), and use them for exact zeros.

Rewriting √72 as the equivalent mixed radical 6√2. Both forms name exactly the same number.

Worked example

Example 5: Radicals and a common trap

Decide whether each pair is equivalent. (a) √12 + √27 and √75 (b) (√2 + √8)² and 18 (c) √(x²) and x

(a) √12 + √27 = 2√3 + 3√3 = 5√3, and √75 = √(25 × 3) = 5√3. Equivalent.
(b) √2 + √8 = √2 + 2√2 = 3√2, and (3√2)² = 9 × 2 = 18. Equivalent.
(c) Try x = −2: √((−2)²) = √4 = 2, but x = −2. Not equivalent. (They agree only for x ≥ 0, because √ always gives a non-negative result.)

(a) yes (b) yes (c) no: x = −2 is a counterexample.

Worked example

Example 6: Three forms of one quadratic

Show that f(x) = (x + 2)² − 1, g(x) = x² + 4x + 3 and h(x) = (x + 1)(x + 3) are the same function, and say what each form tells you about the graph.

Expand f: x² + 4x + 4 − 1 = x² + 4x + 3 = g(x). Expand h: x² + 3x + x + 3 = x² + 4x + 3 = g(x). All three are equivalent.
Vertex form f shows the vertex (−2, −1).
Standard form g shows the y-intercept 3.
Factored form h shows the zeros −1 and −3. You will use these forms constantly in the Quadratic Functions unit.

All three are the same function; each form reveals a different key feature.

Just for fun

Riddle: what is the difference between 2(x + 3) and 2x + 6?

Absolutely nothing, but they still insist on dressing differently.

Hands-on activity

Equivalence card sort

You need: 12 small cards or squares of paper, a pencil and a partner.

Write each of these on its own card: (x + 2)², x² + 4, x² + 4x + 4, 2(x² + 2x + 2) − x², (x − 2)(x + 2), x² − 4, (2 − x)(−x − 2), 3x(x + 1) − 3x, 3x², x(x + 3), x² + 3x, 3x + x².
Shuffle them. Sort them into groups of equivalent expressions by simplifying.
For any card that does not fit a group, find a counterexample that shows it does not match the group you might have expected.

What to look for: Group 1: (x + 2)², x² + 4x + 4, 2(x² + 2x + 2) − x² (all simplify to x² + 4x + 4). Group 2: (x − 2)(x + 2), x² − 4, (2 − x)(−x − 2) (all x² − 4). Group 3: 3x(x + 1) − 3x and 3x² (both 3x²). Group 4: x(x + 3), x² + 3x, 3x + x². The card x² + 4 fits no group; it is not equivalent to (x + 2)²: at x = 1 they give 5 and 9.

Your turn

Are 3(x − 4) + 2x and 5x − 12 equivalent? Show why.

Are (x − 5)² and x² − 25 equivalent? Give a counterexample if not.

Are (x + 4)(x − 1) and x² + 3x − 4 equivalent?

Are √8 + √18 and √50 equivalent?

Are (x² − 9)/(x − 3) and x + 3 equivalent? Explain carefully.

Jamal says 2x² and (2x)² are equivalent. Test x = 3.

Show that √48 and 4√3 are equivalent.

More practice: warm-up

What does "equivalent expressions" mean?

How many counterexamples are needed to show two expressions are not equivalent?

Are x + x and x² equivalent?

Are 4(x + 2) and 4x + 8 equivalent?

Are 1/2 + 1/x and (x + 2)/(2x) equivalent?

More practice: on your own

Are (2x − 1)² − (x + 1)² and 3x(x − 2) equivalent?

Are (x³ − x)/(x² − x) and x + 1 equivalent?

Are (√3 + 1)² and 4 + 2√3 equivalent?

Are 1/(x + 1) + 1/(x − 1) and 2x/(x² − 1) equivalent?

Are (x + 1)³ and x³ + 1 equivalent? Prove your answer.

Challenge

Find the value of k that makes (x + k)² − 9 and (x + 1)(x + 7) equivalent.

Two quadratic expressions agree at x = 0, x = 1 and x = 2. Explain why they must be equivalent.

Is 1/x − 1/(x + 1) equivalent to 1/(x² + x)? State any restrictions.

Check your understanding

Quiz

Which expression is equivalent to (x − 3)²?

Which method can PROVE that two expressions are equivalent?

Which value is a counterexample showing x² ≠ 2x?

Are (x² − 4)/(x + 2) and x − 2 equivalent?

Which pair is NOT equivalent?

Simplify 8x − 2x(4x − 1) − 6.

For parents and teachers

Big idea: Expectation A3.4 asks students to decide whether two algebraic expressions are equivalent by simplifying and by substituting values. This lesson pulls together the polynomial and rational skills from the last four lessons. It also gives a short introduction to A3.2 (√(ab) = √a × √b and equivalent radical forms); the full radicals lesson is "Exact Answers: Simplifying and Operating with Radicals" in the Quadratic Functions unit.

How to help: Write two expressions on paper (for example (x + 2)² and x² + 4) and ask your teen to decide whether they are equivalent and to prove it. Push for a counterexample when they are not, and an algebraic proof when they are.

Common difficulties:

Believing that a few matching values prove equivalence.
Testing only x = 0 or x = 1, which often hide differences.
Ignoring restrictions on rational expressions (equivalent "except at x = ...").
Treating √(x²) as x.
Adding unlike radicals (2√5 + 3√2 is not 5√7) or believing √(a + b) = √a + √b.