Unit 1 · Lesson 5 of 6

Evaluating expressions

About 20 minutes · 5 steps · one idea: substitute inside parentheses

You'll be able to: find the value of an expression or formula when you are told what number each letter stands for, even when the numbers are negative.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first; that effort is the point.

\(\dfrac{4^2 - 1}{\sqrt{9} + 2}\)

The fraction bar groups. Top: \(4^2 = 16\), and \(16 - 1 = 15\). Bottom: \(\sqrt{9} = 3\), and \(3 + 2 = 5\). Divide: \(\dfrac{15}{5} = 3\).

\(\dfrac{6}{7} \cdot \dfrac{14}{15}\)

Cancel 14 and 7 (divide by 7): 2 and 1. Cancel 6 and 15 (divide by 3): 2 and 5. Multiply: \(\dfrac{2 \cdot 2}{1 \cdot 5} = \dfrac{4}{5}\).

Step 2 of 5The idea

To evaluate an expression, replace each variable (letter) with the number you are given, then compute.

A number written right next to a letter means multiply: \(5x\) means \(5 \cdot x\). So if \(x = -2\), then \(5x = 5(-2) = -10\).

Substitute inside parentheses. Write each letter's number inside parentheses, then follow the order of operations from Lesson 5.

Watch the negatives. If \(x = -3\): \(x^2 = (-3)^2 = 9\), but \(-x^2 = -(-3)^2 = -9\).

Two formulas you will meet again: \(b^2 - 4ac\) (Unit 9) and the slope formula \(\dfrac{y_2 - y_1}{x_2 - x_1}\) (Unit 3). The small numbers are labels: \(x_1\) means "the first \(x\)".

Why this shows up on the placement test: many questions say "find the value when \(x = \ldots\)", and slope, the quadratic formula, and functions are all evaluating in disguise.

Step 3 of 5Worked examples

Example 1: evaluate \(2x^2 - 5x + 1\) when \(x = -3\)

  1. Replace each \(x\) with \((-3)\): \(2(-3)^2 - 5(-3) + 1\).
  2. Exponent first: \((-3)^2 = 9\), so it becomes \(2 \cdot 9 - 5(-3) + 1\).
  3. Multiply: \(2 \cdot 9 = 18\) and \(-5(-3) = 15\), so it becomes \(18 + 15 + 1\). (Negative times negative is positive.)
  4. Add: \(34\).

Example 2: evaluate \(b^2 - 4ac\) when \(a = 2\), \(b = -3\), \(c = -5\)

  1. Substitute: \((-3)^2 - 4(2)(-5)\).
  2. Exponent: \((-3)^2 = 9\). Multiply: \(4(2)(-5) = 8(-5) = -40\).
  3. Subtract: \(9 - (-40) = 9 + 40 = 49\). (Subtracting a negative means adding.)

Step 4 of 5Practice

Substitute with parentheses on paper, then type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.

Evaluate \(3x - 7\) when \(x = -2\).

\(3(-2) - 7 = -6 - 7 = -13\).

Evaluate \(x^2 + 4x\) when \(x = -5\).

\((-5)^2 + 4(-5) = 25 + (-20) = 5\).

Evaluate \(-x^2 + 2x\) when \(x = -4\).

\(-(-4)^2 + 2(-4)\). The exponent comes first: \((-4)^2 = 16\), so the first term is \(-16\). Then \(2(-4) = -8\). Total: \(-16 + (-8) = -24\).

Evaluate \(\dfrac{a + b}{a - b}\) when \(a = 2\) and \(b = -6\).

Top: \(2 + (-6) = -4\). Bottom: \(2 - (-6) = 2 + 6 = 8\). Divide: \(\dfrac{-4}{8} = -\dfrac{1}{2}\).

Evaluate \(b^2 - 4ac\) when \(a = 3\), \(b = -2\), and \(c = 5\).

\((-2)^2 - 4(3)(5) = 4 - 60 = -56\).

Evaluate the slope formula \(\dfrac{y_2 - y_1}{x_2 - x_1}\) when \(x_1 = -1\), \(y_1 = 4\), \(x_2 = 3\), and \(y_2 = -2\).

Top: \(-2 - 4 = -6\). Bottom: \(3 - (-1) = 3 + 1 = 4\). Divide and simplify: \(\dfrac{-6}{4} = -\dfrac{3}{2}\).

The formula \(F = \dfrac{9}{5}C + 32\) changes a Celsius temperature \(C\) into Fahrenheit \(F\). What is \(F\) when \(C = -10\)?

\(F = \dfrac{9}{5}(-10) + 32\). Multiply first: \(\dfrac{9}{5} \cdot \dfrac{-10}{1} = \dfrac{9 \cdot (-2)}{1} = -18\) (cancel 5 and 10). Then \(-18 + 32 = 14\). So it is \(14^\circ\) F.

If \(x = -2\) and \(y = 3\), what is the value of \(x^2 y - xy\)?

  1. \(-18\)
  2. \(6\)
  3. \(18\)
  4. \(-6\)

Substitute: \((-2)^2(3) - (-2)(3)\). First part: \(4 \cdot 3 = 12\). Second part: \((-2)(3) = -6\).

Then \(12 - (-6) = 12 + 6 = 18\). Answer: (c).

(b) forgets that \(xy\) is negative. (d) treats \((-2)^2\) as \(-4\). (a) makes both slips.

Step 5 of 5Wrap-up

You can now evaluate expressions and formulas by putting each number in parentheses first and then following the order of operations.

Read or watch next: OpenStax Elementary Algebra 2e, 1.2 Use the Language of Algebra (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Substitution and evaluating expressions (short videos). Quick rules: Number skills cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "When I plug in a negative number, when do I need parentheses?" It knows this course and can explain things another way.