--- title: "Locating Roots and Iterative Methods" description: "Use continuity and opposite signs to locate a root, rearrange an equation as x = g(x), and calculate successive estimates. Check a justified error bound or a sign-change bracket before reporting the r" canonical: https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM source: https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Locating Roots and Iterative Methods Use continuity and opposite signs to locate a root, rearrange an equation as x = g(x), and calculate successive estimates. Check a justified error bound or a sign-change bracket before reporting the required accuracy. Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM ## Ready when they can - Use a sign change to show a root lies in a given interval - Rearrange an equation into an iterative formula - Run an iteration to a stated accuracy and justify stopping ## Lesson: Trapping a root, then walking in A root is an input x where f(x) = 0. To evaluate f(a), replace every x in the rule with a, then calculate powers before multiplication and subtraction. A function is continuous on an interval if its graph has no break there. Opposite endpoint signs then guarantee at least one root strictly between the endpoints. If an endpoint value is zero, that endpoint is itself a root. **Example.** For f(x) = x³ - x - 3, f(1) = 1 - 1 - 3 = -3 and f(2) = 8 - 2 - 3 = 3. This polynomial is continuous, so a root lies between 1 and 2. Rearrange x³ = x + 3 as x = the cube root of (x + 3). Starting at 1.5, repeatedly apply this rule to the latest value. Keep the full calculator value between updates; the next picture shows rounded displays. *(drawing: These estimates suggest a location. A later sign check will certify the rounded answer.)* Opposite signs do not prove there is only one root. Same signs do not rule out roots: (x - 3)² is positive at 2 and 4 but zero at 3. Continuity matters too. The function 1/(x - 2) changes sign between 1 and 3, but is undefined at 2 and never equals zero. When searching a table, compare signs at neighbouring inputs to find a bracket. To form x = g(x), use equivalent algebra and keep any restrictions. For x³ + 2x = 9, subtract 2x and take cube roots: x = the cube root of (9 - 2x). For x² = 2x + 8, divide every term by x: x = 2 + 8/x. This rule requires a nonzero input. Check that each proposed starting value is allowed. Write an update as x_next = g(x). The notation x0, also written x_0, names the starting value. Then x1 is the first output and x2 is the second. Always use the latest output as the next input. For x_next = (x + 4)/2, starting at x0 = 0 gives x1 = 2, x2 = 3 and x3 = 3.5. Keep calculator precision between updates and round only the reported answer. A fixed point r satisfies r = g(r). For g(x) = (x + 4)/2, 2r = r + 4 gives r = 4. A valid rearrangement need not converge. For a linear rule g(x) = mx + b, subtract r = mr + b: g(x) - r = m(x - r). Each absolute error is multiplied by |m|. It shrinks when |m| < 1 and grows when |m| > 1, unless the start is r. Absolute error is |estimate - root|. If an initial error bound is B and each update multiplies errors by c, where 0 < c < 1, multiply the bound by c after each update. Stop when it is below the requested tolerance. Here 1-decimal accuracy means error below 0.05; 2-decimal accuracy means below 0.005. A sign bracket's midpoint is at most half its width from a root inside it. Correct rounding also needs a bracket within one rounding interval. Suppose a root is bracketed between 4.314 and 4.316. The interval crosses 4.315, the boundary between rounding to 4.31 and rounding to 4.32. Values inside the bracket can round either way. Refine the bracket before reporting the root to two decimal places. **Example.** Return to f(x) = x³ - x - 3. Calculate f(1.671) = -0.005165289 and f(1.672) = 0.004216448. Continuity and opposite signs place a root between 1.671 and 1.672. Every value in this bracket rounds to 1.67, so the sign check certifies the two-decimal answer suggested by iteration. Matching rounded iterates do not certify the root. For x_next = 0.9999x + 0.0001, the fixed point is 1. Starting at zero gives 0.0001 and then 0.00019999. Both round to 0.000 at three decimals, although both are far from 1. Use a justified error bound or a continuous-function sign bracket that certifies the requested accuracy. A small function value is not an x-error bound either. For f(x) = 0.001(x - 10), the root is 10. At x = 9, f(9) = -0.001, but the estimate is still 1 away from the root. The function's scale affects its values. Check the distance using a justified bound or bracket. **Recap.** Locate a root using continuity and opposite signs. Feed each output into x = g(x). Certify the required accuracy with a justified error bound or sign bracket. *This lesson, question bank and flashcards were written for this dataset.* ## Practice 18 questions on this page, each with its working shown. ## Optional connections - [Recognising cubic, reciprocal and exponential graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9) ## Opens up - [The Newton-Raphson Method](https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd) - [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV)