Grade by Grade
C. Polynomial and Rational Functions · Rational Functions and Equations

Rational Equations with Quadratics, Formulas and Inverses

C3.6 C3.7About 70 minutes
Learning goals
  • I can solve rational equations with a rational expression on each side, including ones that need the quadratic formula.
  • I can recognize and explain when a rational equation has no real solutions.
  • I can rearrange formulas that contain reciprocals, such as the thin-lens equation and the formula for resistors in parallel.
  • I can find the inverse of y = (ax + b)/(cx + d) by solving a rational equation for x.

Words to know

literal equation
An equation with several letters, such as 1/f = 1/d_o + 1/d_i. Solving it means isolating one chosen letter.
thin-lens equation
1/f = 1/d_o + 1/d_i, linking the focal length f of a lens, the object distance d_o and the image distance d_i.
resistors in parallel
Two resistors side by side in a circuit. Their combined resistance R satisfies 1/R = 1/R₁ + 1/R₂.
self-inverse function
A function that is its own inverse: f⁻¹(x) = f(x). Its graph is symmetric about the line y = x.

Equations that model real things

Photographers, eye doctors and electricians all use formulas full of reciprocals. A camera lens obeys 1/f = 1/d_o + 1/d_i. Resistors side by side in a circuit obey 1/R = 1/R₁ + 1/R₂. Solving these formulas is exactly the skill from the last lesson: clear the fractions, solve, check. This lesson adds three twists: equations that turn into quadratics needing the quadratic formula, equations with no real solution, and formulas where you solve for a letter instead of a number.

Rational expressions on both sides

When each side is a single fraction, cross-multiplying is quickest. The x² terms often cancel, leaving a linear equation, but not always.

If you end up with a quadratic that does not factor, use the quadratic formula x = (−b ± √(b² − 4ac))/(2a), which you used in the MCR3U unit Quadratic Functions. If its discriminant is negative, the equation has no real roots, and the two graphs never meet.

Worked example

Example 1: The x² terms cancel

Solve (x + 1)/(x − 2) = (x − 3)/(x + 4).

  1. Restrictions: x ≠ 2, x ≠ −4.
  2. Cross-multiply: (x + 1)(x + 4) = (x − 3)(x − 2).
  3. x² + 5x + 4 = x² − 5x + 6. The x² terms cancel: 10x = 2, so x = 1/5.
  4. Check: (1.2)/(−1.8) = −2/3 and (−2.8)/(4.2) = −2/3 ✓.

x = 1/5

Worked example

Example 2: No real solution

Solve 2x/(x − 1) = (x + 3)/(x + 1).

  1. Restrictions: x ≠ 1, x ≠ −1.
  2. Cross-multiply: 2x(x + 1) = (x + 3)(x − 1).
  3. 2x² + 2x = x² + 2x − 3, so x² = −3.
  4. No real number squares to a negative number.

No real solution. The graphs of the two sides never intersect (see below).

y = 2x/(x − 1) and y = (x + 3)/(x + 1) never meetxy−5−4−3−2−112345−5−4−3−2−1123456x = 1x = −1y = 2x/(x − 1)y = (x + 3)/(x + 1)
y = 2x/(x − 1) (solid, asymptotes x = 1 and y = 2) and y = (x + 3)/(x + 1) (dashed, asymptotes x = −1 and y = 1). They come close but never cross, which matches x² = −3 having no real root.
Worked example

Example 3: The quadratic formula

Solve 1/x + 2/(x + 1) = 1. Give exact answers and decimals.

  1. Restrictions: x ≠ 0, x ≠ −1. LCD x(x + 1).
  2. Multiply: (x + 1) + 2x = x(x + 1), so 3x + 1 = x² + x.
  3. x² − 2x − 1 = 0. This does not factor, so x = (2 ± √(4 + 4))/2 = (2 ± 2√2)/2 = 1 ± √2.
  4. Decimals: about 2.414 and −0.414. Neither is restricted.
  5. Check x ≈ 2.414: 1/2.414 + 2/3.414 ≈ 0.414 + 0.586 = 1 ✓.

x = 1 ± √2 (about 2.41 and −0.41).

Formulas with reciprocals

To isolate one letter in a formula like 1/f = 1/d_o + 1/d_i, treat the other letters like numbers. Find the LCD, multiply every term by it, collect the terms containing your chosen letter on one side, and factor that letter out.

Worked example

Example 4: The thin-lens equation

Solve 1/f = 1/d_o + 1/d_i for d_i. Then find the image distance for a lens with focal length 10 cm and an object 30 cm away.

  1. Subtract: 1/d_i = 1/f − 1/d_o.
  2. Common denominator: 1/d_i = (d_o − f)/(f·d_o).
  3. Take reciprocals of both sides: d_i = f·d_o/(d_o − f).
  4. Substitute: d_i = (10)(30)/(30 − 10) = 300/20 = 15.
  5. Check: 1/30 + 1/15 = 1/30 + 2/30 = 3/30 = 1/10 ✓.

d_i = f·d_o/(d_o − f); the image forms 15 cm behind the lens.

FFobjectimaged_o = 30 cmd_i = 15 cmf = 10 cm
Ray diagram for f = 10 cm and an object 30 cm from the lens, drawn to scale. The three standard rays meet 15 cm behind the lens, where an upside-down image half the size of the object forms, just as the formula predicts.
Worked example

Example 5: Resistors in parallel

Two resistors in parallel have combined resistance R given by 1/R = 1/R₁ + 1/R₂. (a) Solve for R. (b) If R = 4 Ω and R₁ = 6 Ω, find R₂.

  1. (a) Common denominator: 1/R = (R₂ + R₁)/(R₁R₂). Take reciprocals: R = R₁R₂/(R₁ + R₂).
  2. (b) 1/R₂ = 1/R − 1/R₁ = 1/4 − 1/6 = 3/12 − 2/12 = 1/12.
  3. So R₂ = 12 Ω. Check: 6 × 12/(6 + 12) = 72/18 = 4 ✓.

(a) R = R₁R₂/(R₁ + R₂) (b) R₂ = 12 Ω

Worked example

Example 6: The inverse of a rational function

Find the inverse of f(x) = (2x + 1)/(x − 3).

  1. Write y = (2x + 1)/(x − 3) and swap x and y: x = (2y + 1)/(y − 3).
  2. Multiply by y − 3: xy − 3x = 2y + 1.
  3. Collect the y-terms: xy − 2y = 3x + 1, so y(x − 2) = 3x + 1.
  4. Divide: y = (3x + 1)/(x − 2).

f⁻¹(x) = (3x + 1)/(x − 2). The asymptotes swap: f has x = 3 and y = 2; f⁻¹ has x = 2 and y = 3. The domain of f is the range of f⁻¹, and the reverse.

f(x) = (2x + 1)/(x − 3) and its inversexy−4−22468−6−4−22468x = 3x = 2y = 2y = 3ff⁻¹y = x
f(x) = (2x + 1)/(x − 3) and f⁻¹(x) = (3x + 1)/(x − 2) are reflections of each other in the line y = x (dashed). Each vertical asymptote of one is a horizontal asymptote of the other.
Just for fun

Why did the optometrist love rational equations? Everything came into focus as soon as the denominators were cleared.

Hands-on activity

Measure the focal length of a magnifying glass

You need: a magnifying glass, a lamp with the shade removed or a bright window, a sheet of white paper taped to a wall or a box, and a measuring tape. Grown-up note: never point the magnifying glass at the sun or use it to focus sunlight on anything. Focused sunlight can start a fire and can permanently damage eyes. Do this indoors, with a lamp or a window that is not in direct sunlight. A hot lamp bulb can burn, so do not touch it.

  1. Far object. Stand across the room from the window. Move the magnifying glass between the window and the paper until you see a sharp, upside-down image of the window. Measure the lens-to-paper distance. For a very far object, 1/d_o is almost 0, so this distance is about f.
  2. Near object. Put the lamp about 1 m (d_o = 100 cm) from the lens. Move the paper until the lamp's image is sharp, and measure d_i.
  3. Use 1/f = 1/d_o + 1/d_i, solved as f = d_o·d_i/(d_o + d_i), to calculate f from your step 2 measurements. Compare it with step 1.
  4. Predict d_i for d_o = 60 cm using d_i = f·d_o/(d_o − f). Test your prediction.

What to look for: your numbers depend on your lens. For example, if f = 20 cm, the lamp at 100 cm gives d_i = 2000/80 = 25 cm, and at 60 cm it gives 1200/40 = 30 cm. Your two values of f should agree within a few centimetres; the image is only sharp over a small range, which limits the precision.

Your turn

1

Solve (x + 2)/(x − 1) = (x + 5)/(x + 3).

2

Solve 1/(x + 1) + 1/(x − 1) = 1. Give exact answers.

3

Solve x/(x + 1) = (x + 2)/(x + 4).

4

Solve 1/f = 1/a + 1/b for f.

5

A lens has focal length 20 cm and an object is 60 cm away. Find the image distance.

6

Find the combined resistance of 3 Ω and 6 Ω resistors in parallel.

7

Find the inverse of f(x) = (x + 4)/(x − 1). What do you notice?

More practice: warm-up

1

Solve 3/x = x/12.

2

Solve 1/R = 1/2 + 1/2.

3

Solve 2/(x − 1) = 1/(x + 1).

4

Solve v = d/t for t.

5

Solve 1/x = 1/3 + 1/6.

More practice: on your own

1

Solve (2x − 1)/(x + 2) = (x + 1)/(x − 1). Give exact answers.

2

Solve x/(x − 2) = (x − 4)/(x + 3).

3

Solve a = (b + 2)/(b − 1) for b.

4

Show that x + 4/x = 3 has no real solutions.

5

An object is 15 cm from a lens with focal length 10 cm. Where does the image form?

Challenge

1

Two resistors in parallel have a combined resistance of 6 Ω. One resistor is 5 Ω more than the other. Find both.

2

Show that f(x) = (ax + b)/(cx − a) is its own inverse (for any a, b, c with a² + bc ≠ 0).

3

A lens has focal length 12 cm. Where must an object be placed so that the image distance is twice the object distance?

Check your understanding

Quiz

1.

What is the solution of (x + 1)/(x − 2) = (x − 3)/(x + 4)?

2.

How many real solutions does 2x/(x − 1) = (x + 3)/(x + 1) have?

3.

Which is 1/f = 1/d_o + 1/d_i solved for d_i?

4.

What is the combined resistance of a 6 Ω and a 3 Ω resistor in parallel?

5.

What is the inverse of f(x) = (2x + 1)/(x − 3)?

6.

How many real solutions does x + 4/x = 3 have?

For parents and teachers

Big idea (C3.6, C3.7): the same 'clear the fractions' method solves equations with fractions on both sides, equations that lead to quadratics, and formulas with reciprocals (lenses, circuits). A negative discriminant means no real solution, and the graphs then never meet.

How to help: The magnifying-glass activity is a real physics measurement; please supervise and keep it away from sunlight. Ask your teen to show you how they solved the lens formula for d_i, and to check it by substituting numbers.

Common difficulties:

  • 'Taking the reciprocal' of a sum term by term: 1/(a + b) is not 1/a + 1/b.
  • Losing a sign when moving terms in literal equations.
  • Confusing the inverse f⁻¹(x) with the reciprocal 1/f(x).
  • Stopping at x² = −3 and writing x = ±√3 (the square root of a negative number is not real).