Roots Discovery

Numerical Roots Sequence // Lesson 1

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1 Visualizing the Unknown

Consider the function \( f(x) = x^2 - 2 \). A root is where \( f(x) = 0 \). While we know the exact value is \( \sqrt{2} \), a computer sees this as a point on a coordinate plane.

Graph Area

Task 1.1

Sketch the graph of \( f(x) = x^2 - 2 \). Circle the locations where the graph crosses the x-axis.

Task 1.2

Estimate the value of the positive root based on your sketch. How accurate can you be just by looking?

2 Hitting the Algebraic Wall

We have the Quadratic Formula for degree-2 polynomials. There is a cubic formula and a quartic formula (though they are massive). But for anything degree 5 or higher, a proof exists showing no general algebraic formula can ever exist.

Equation A: \( x^2 - 5x + 6 = 0 \)

Solve using factoring or the quadratic formula. Show your work below.

Equation B: \( x^5 - x - 1 = 0 \)

Try to isolate \( x \) using standard algebra. What happens? Why can't we solve this the usual way?

3 Thinking Like a Computer

If you can't solve it exactly, you have to hunt for it. This is called a Numerical Method.

// Algorithm: The Bisection Method (A simple hunt)

Pick a range [a, b] where the root must live.
Find the midpoint m = (a + b) / 2.
Check if the root is in [a, m] or [m, b].
Discard the half without the root.
Repeat until the range is tiny.

Let's use the Bisection Method to find \( \sqrt{7} \) (the root of \( x^2 - 7 = 0 \)). We know \( \sqrt{7} \) is between 2 and 3.

Iteration	Range [a, b]	Midpoint (m)	Result of \( m^2 - 7 \)
1	[2, 3]	2.5	\( 2.5^2 - 7 = -0.75 \) (Low)
2	[2.5, 3]	2.75	\( 2.75^2 - 7 = 0.56 \) (High)
3	[2.5, 2.75]	2.625
4

Reflection:

The Bisection Method works, but it's slow. After 2 iterations, we only narrowed it down by about 0.25. How many steps do you think it would take to get to 10 decimal places? Why might we need a faster method?

Numerical Roots Sequence

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Tangent Path Lesson

Manual Iterations Lesson

Convergence Simulation Lesson

Computational Error Lesson

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Convergence Tracker Project Guide

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Roots Exit Ticket

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