Exercise 5.4 Class 10 Math Solutions – Equations Reducible to Quadratic Form, Punjab Board

Solved Exercise 5.4 for Class 10 Mathematics, Punjab Board. Comprehensive step-by-step notes covering equations reducible to quadratic equations using algebraic substitution, solving auxiliary quadratic equations, back-substitution, and solution sets.

This page provides complete solved solutions for Exercise 5.4 from Unit 5: Algebraic Functions, Class 10 Mathematics, Punjab Board (PCTB). It covers the solution of equations reducible to quadratic equations. Students learn algebraic substitution techniques to transform higher-degree, reciprocal, and fractional equations into standard quadratic equations, solve auxiliary equations, back-substitute to find original roots, and write final solution sets for Matric Part 2 exams.
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Exercise Overview

Exercise 5.4 in Unit 5: Algebraic Functions of Class 10 Punjab Board Mathematics introduces students to the powerful technique of solving equations reducible to quadratic form. Many complex algebraic expressions that initially appear quartic, reciprocal, or fractional can be simplified into standard quadratic form ax² + bx + c = 0 through appropriate variable substitutions.

Throughout this exercise, students learn to recognize recurring algebraic patterns, assign an auxiliary variable (such as y = x² or y = x + 1/x), and rewrite the problem as a quadratic equation in y. After determining the roots of the auxiliary quadratic equation using factorization or the quadratic formula, students perform back-substitution to solve for the original variable x. The exercise also highlights the necessity of verifying solutions to detect any extraneous roots arising from fractional or radical terms.

In Punjab Board examinations (including BISE Lahore, Rawalpindi, Faisalabad, Gujranwala, Multan, Sargodha, Bahawalpur, and DG Khan), questions from Exercise 5.4 are prominent candidates for Section C long questions and detailed 4-mark short questions. Mastering the systematic substitution and back-substitution workflow presented in these verified notes ensures maximum score attainment.

FAQs

Are these Exercise 5.4 solutions according to the Punjab Board syllabus?
Yes. These solutions follow the Punjab Curriculum and Textbook Board (PCTB) syllabus for Class 10 Mathematics, covering all Unit 5 Exercise 5.4 reducible quadratic equation types for the 2026-2027 session.
What is an equation reducible to quadratic form?
An equation reducible to quadratic form is a non-quadratic algebraic equation that can be converted into standard quadratic form ax² + bx + c = 0 by a suitable substitution variable (such as y = x² or y = x + 1/x).
What is the general method to solve equations reducible to quadratic form in Exercise 5.4?
1) Identify the repeated variable expression and set y = f(x), 2) rewrite the equation in terms of y, 3) solve for y, 4) back-substitute to find x, and 5) verify the solution set.
What types of reducible equations are covered in Exercise 5.4?
Key types include equations involving higher even powers (like ax⁴ + bx² + c = 0), reciprocal equations (x + 1/x = y), exponential equations, and radical expressions reducible to quadratic equations.
Why is back-substitution required when solving reducible equations?
Back-substitution is essential because the substitution variable y is only an auxiliary tool; replacing y with the original expression f(x) gives the actual roots for the original variable x.
Do students need to check for extraneous roots in Exercise 5.4?
Yes, especially in rational and radical equations, substituting candidate roots into the original equation ensures that invalid or extraneous roots are discarded from the final solution set.
Are all questions in Exercise 5.4 solved with complete working steps?
Yes, every problem is solved step-by-step with explicit substitutions, auxiliary quadratic solutions via factoring or the quadratic formula, and detailed back-substitution.
Can students read these Exercise 5.4 solutions online?
Yes, the complete solved PDF notes for Exercise 5.4 are embedded to read online directly on MaryamNotes.pk with no compulsory download.
Are these Exercise 5.4 solutions free for students?
Yes, all Exercise 5.4 solutions on MaryamNotes.pk are 100% free to read online for all Class 10 students across all Punjab boards.