Math Problem Statement

Solve the cubic equations: 1) f(x) = x^3 - 2x^2 - 5x + 6 = 0, and 2) 2x^3 - 20x^2 - 618x + 1980 = 0.

Solution

The image contains two problems involving cubic equations (third-degree equations) to solve and verify. Let's break them down:

Problem 1:

Given the function f(x)=x32x25x+6f(x) = x^3 - 2x^2 - 5x + 6, the goal is to solve the equation f(x)=0f(x) = 0.

Steps:

  1. a) Verify that x=1x = 1 is a solution of the equation f(x)=0f(x) = 0.

    This means substituting x=1x = 1 into the equation f(x)f(x) and checking if the result is 0.

  2. b) Show that the cubic function can be written in the form (x1)(ax2+bx+c)(x - 1)(ax^2 + bx + c) by expanding and identifying the coefficients of aa, bb, and cc.

  3. c) Solve the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.

  4. d) Deduce all the solutions of the equation f(x)=0f(x) = 0, by factoring it completely.

Problem 2:

Solve the equation 2x320x2618x+1980=02x^3 - 20x^2 - 618x + 1980 = 0.

Steps:

  1. a) Verify that x=3x = 3 is a solution of this equation. Then factor the cubic polynomial by expressing it as (x3)g(x)(x - 3)g(x), where g(x)g(x) is a quadratic polynomial.

  2. b) Find all the solutions of the equation by solving the quadratic polynomial g(x)=0g(x) = 0.


Do you want me to walk you through each step in detail, or would you prefer specific assistance with any part of these problems?

Relative Questions:

  1. How do we confirm whether a given value is a solution to a cubic equation?
  2. What is the process for factoring a cubic polynomial?
  3. How can we determine the coefficients when rewriting a cubic equation as a product of a linear and quadratic term?
  4. What methods can we use to solve quadratic equations?
  5. How can we use factoring to find all solutions of a cubic equation?

Tip:

When verifying if a specific value is a solution, always substitute the value into the equation and simplify to check if the result is zero.

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Math Problem Analysis

Mathematical Concepts

Cubic Equations
Factoring
Quadratic Equations
Verification of Roots

Formulas

Cubic equation: f(x) = ax^3 + bx^2 + cx + d = 0
Factoring: (x - root)(quadratic polynomial)
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Theorems

Rational Root Theorem
Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12