UNIT 1 • FINDING ROOTS OF EQUATIONS

The Bisection Method
Step-by-Step Friendly Guide

The Bisection Method is a straightforward and reliable numerical technique used to find roots (solutions) of continuous functions. It works by repeatedly dividing an interval in half and selecting the subinterval where the function changes sign, thereby narrowing down the location of the root.

HOW THE BISECTION METHOD WORKSMIDPOINT CALCULATION

C = \frac{a + b}{2}

1. Select Interval [a, b]Choose two points

a

and

b

such that

f(a)

and

f(b)

have opposite signs.

2. Calculate Midpoint (C)Compute

C = \frac{a+b}{2}

to split interval in half.

3. Evaluate f(C)Check sign and value of

f(C)

4. Update IntervalIf

f(C) \approx 0

C

is the root! Otherwise, set

a=C

b=C

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find a root of $x^3 - 4x - 9 = 0$ correct to one decimal place (tolerance $\varepsilon = 0.05$ ) using the Bisection Method.

STEP 01 • REARRANGING THE EQUATION

First, set the equation to zero to get our standard function $f(x) = 0$ :

f(x) = x^3 - 4x - 9 = 0

STEP 02 • CHOOSE INITIAL POINTS

Choose two initial points $a$ and $b$ such that $f(a)$ and $f(b)$ have opposite signs. Let's choose $a = 2$ and $b = 3$ :

f(2) = 2^3 - 4(2) - 9 = 8 - 8 - 9 = -9 \quad (\text{Negative})

f(3) = 3^3 - 4(3) - 9 = 27 - 12 - 9 = 6 \quad (\text{Positive})

Since $f(2) < 0$ and $f(3) > 0$ , a root must lie in $[2, 3]$ . Each step halves the interval, so the number of steps is known in advance: stop at step $k$ once $\frac{b-a}{2^k} < \varepsilon$ , i.e. $k > \log_2 \frac{3-2}{0.05} = \log_2 20 \approx 4.32$ , so 5 steps.

STEP 03 • CALCULATE FIRST MIDPOINT

Calculate the midpoint $C$ of the interval $[a, b] = [2, 3]$ :

C = \frac{a + b}{2} = \frac{2 + 3}{2} = 2.5

f(2.5) = (2.5)^3 - 4(2.5) - 9 = 15.625 - 10 - 9 = -3.375 \quad (\text{Negative})

STEP 04 • UPDATE THE INTERVAL

Since $f(2) < 0$ and $f(2.5) < 0$ , both left points are negative. The sign change happens between $C = 2.5$ and $b = 3$ .

Set new left bound:

a = C = 2.5

. New interval is

[2.5, 3.0]

STEP 05 • STEP-BY-STEP ITERATIONS WALKTHROUGH

Iteration 2:

Midpoint:

C_2 = \frac{2.5 + 3}{2} = 2.75

Function Value:

f(2.75) = (2.75)^3 - 4(2.75) - 9 = +0.7969

Update:

f(2.5) < 0

and

f(2.75) > 0

, so set

b = 2.75

. New interval:

[2.5, 2.75]

Iteration 3:

Midpoint:

C_3 = \frac{2.5 + 2.75}{2} = 2.625

Function Value:

f(2.625) = -1.4121

Update: set

a = 2.625

. New interval:

[2.625, 2.75]

Iteration 4:

Midpoint:

C_4 = \frac{2.625 + 2.75}{2} = 2.6875

Function Value:

f(2.6875) = -0.3391

Update: set

a = 2.6875

. New interval:

[2.6875, 2.75]

Iteration 5:

Midpoint:

C_5 = \frac{2.6875 + 2.75}{2} = 2.71875

Function Value:

f(2.71875) = +0.2209

Check:

\frac{b - a}{2} = \frac{2.75 - 2.6875}{2} = 0.03125 < 0.05

, so stop. The root lies within 0.03 of 2.71875.

STEP 06 • ITERATION SUMMARY TABLE

Step (k)	Left Bound (a)	Right Bound (b)	Midpoint (C)	Value f(C)	Sign of f(C)	Next Interval
1	2.0000	3.0000	2.5000	-3.3750	f(2.5) < 0	[2.5000, 3.0000]
2	2.5000	3.0000	2.7500	+0.7969	f(2.75) > 0	[2.5000, 2.7500]
3	2.5000	2.7500	2.6250	-1.4121	f(2.625) < 0	[2.6250, 2.7500]
4	2.6250	2.7500	2.6875	-0.3391	f(2.6875) < 0	[2.6875, 2.7500]
5	2.6875	2.7500	2.7188	+0.2209	(b − a)/2 = 0.0313 < 0.05	Stop

FINAL CONCLUSION

Root Correct to 1 Decimal Place: x ≈ 2.7

x \approx 2.71875 \Rightarrow x = 2.7 \quad (\text{true root } 2.7065)

The Bisection Method guarantees convergence as long as you start with points that bracket the root. Notice how the interval width $|b - a|$ is cut in half at every step ( $1 \to 0.5 \to 0.25 \to 0.125 \to 0.0625$ ). After 5 steps the half-width is below $0.05$ , giving the root $x \approx 2.7$ . Each extra decimal place costs about 3.3 more steps (4 decimal places need 14 steps).

LABORATORY

Interactive Live Calculator & Visualizer

ROOT FINDER LABORATORY

Bisection Method Interactive Engine

Unit 1•1 of 63

NextSecant Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 1 • FINDING ROOTS OF EQUATIONS

The Bisection Method
Step-by-Step Friendly Guide

HOW THE BISECTION METHOD WORKSMIDPOINT CALCULATION

C = \frac{a + b}{2}

1. Select Interval [a, b]Choose two points

a

and

b

such that

f(a)

and

f(b)

have opposite signs.

2. Calculate Midpoint (C)Compute

C = \frac{a+b}{2}

to split interval in half.

3. Evaluate f(C)Check sign and value of

f(C)

4. Update IntervalIf

f(C) \approx 0

C

is the root! Otherwise, set

a=C

b=C

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find a root of $x^3 - 4x - 9 = 0$ correct to one decimal place (tolerance $\varepsilon = 0.05$ ) using the Bisection Method.

STEP 01 • REARRANGING THE EQUATION

First, set the equation to zero to get our standard function $f(x) = 0$ :

f(x) = x^3 - 4x - 9 = 0

STEP 02 • CHOOSE INITIAL POINTS

Choose two initial points $a$ and $b$ such that $f(a)$ and $f(b)$ have opposite signs. Let's choose $a = 2$ and $b = 3$ :

f(2) = 2^3 - 4(2) - 9 = 8 - 8 - 9 = -9 \quad (\text{Negative})

f(3) = 3^3 - 4(3) - 9 = 27 - 12 - 9 = 6 \quad (\text{Positive})

STEP 03 • CALCULATE FIRST MIDPOINT

Calculate the midpoint $C$ of the interval $[a, b] = [2, 3]$ :

C = \frac{a + b}{2} = \frac{2 + 3}{2} = 2.5

f(2.5) = (2.5)^3 - 4(2.5) - 9 = 15.625 - 10 - 9 = -3.375 \quad (\text{Negative})

STEP 04 • UPDATE THE INTERVAL

Since $f(2) < 0$ and $f(2.5) < 0$ , both left points are negative. The sign change happens between $C = 2.5$ and $b = 3$ .

Set new left bound:

a = C = 2.5

. New interval is

[2.5, 3.0]

STEP 05 • STEP-BY-STEP ITERATIONS WALKTHROUGH

Iteration 2:

Midpoint:

C_2 = \frac{2.5 + 3}{2} = 2.75

Function Value:

f(2.75) = (2.75)^3 - 4(2.75) - 9 = +0.7969

Update:

f(2.5) < 0

and

f(2.75) > 0

, so set

b = 2.75

. New interval:

[2.5, 2.75]

Iteration 3:

Midpoint:

C_3 = \frac{2.5 + 2.75}{2} = 2.625

Function Value:

f(2.625) = -1.4121

Update: set

a = 2.625

. New interval:

[2.625, 2.75]

Iteration 4:

Midpoint:

C_4 = \frac{2.625 + 2.75}{2} = 2.6875

Function Value:

f(2.6875) = -0.3391

Update: set

a = 2.6875

. New interval:

[2.6875, 2.75]

Iteration 5:

Midpoint:

C_5 = \frac{2.6875 + 2.75}{2} = 2.71875

Function Value:

f(2.71875) = +0.2209

Check:

\frac{b - a}{2} = \frac{2.75 - 2.6875}{2} = 0.03125 < 0.05

, so stop. The root lies within 0.03 of 2.71875.

STEP 06 • ITERATION SUMMARY TABLE

Step (k)	Left Bound (a)	Right Bound (b)	Midpoint (C)	Value f(C)	Sign of f(C)	Next Interval
1	2.0000	3.0000	2.5000	-3.3750	f(2.5) < 0	[2.5000, 3.0000]
2	2.5000	3.0000	2.7500	+0.7969	f(2.75) > 0	[2.5000, 2.7500]
3	2.5000	2.7500	2.6250	-1.4121	f(2.625) < 0	[2.6250, 2.7500]
4	2.6250	2.7500	2.6875	-0.3391	f(2.6875) < 0	[2.6875, 2.7500]
5	2.6875	2.7500	2.7188	+0.2209	(b − a)/2 = 0.0313 < 0.05	Stop

FINAL CONCLUSION

Root Correct to 1 Decimal Place: x ≈ 2.7

x \approx 2.71875 \Rightarrow x = 2.7 \quad (\text{true root } 2.7065)

LABORATORY

Interactive Live Calculator & Visualizer

ROOT FINDER LABORATORY

Bisection Method Interactive Engine

Unit 1•1 of 63

NextSecant Method

The Bisection Method Step-by-Step Friendly Guide

Step-by-Step Worked Example

Root Correct to 1 Decimal Place: x ≈ 2.7

Interactive Live Calculator & Visualizer

Bisection Method Interactive Engine

NETZ Engine

The Bisection Method Step-by-Step Friendly Guide

Step-by-Step Worked Example

Root Correct to 1 Decimal Place: x ≈ 2.7

Interactive Live Calculator & Visualizer

Bisection Method Interactive Engine

The Bisection Method
Step-by-Step Friendly Guide

The Bisection Method
Step-by-Step Friendly Guide