EQUATIONS IN QUADRATIC FORM || GRADE 9 MATHEMATICS Q1

Summary

In this educational video, viewers are guided through solving various types of quadratic and rational algebraic equations. The tutorial covers transforming equations into quadratic form, simplifying rational expressions, and solving both standard and non-standard quadratic equations using factoring. Key steps are demonstrated with clear examples, including how to find least common denominators (LCD) and apply them to solve equations. The video concludes with a solution check, ensuring viewers understand how to verify their results. This comprehensive approach helps learners grasp foundational concepts in algebraic problem-solving.

Takeaways

• 😀 Learn how to simplify algebraic expressions involving rational terms, such as 1/x + 2x/5, by finding the least common denominator (LCD) and multiplying both sides by it.
• 😀 Understand the concept of multiplying terms involving fractions, canceling common factors, and simplifying to obtain expressions like x^2 + 5/5x.
• 😀 Grasp how to combine terms in algebraic fractions, for example, x/(x+1) + 2/(x+2), and solve by multiplying by the LCD.
• 😀 See how to expand quadratic equations that are not initially in standard form, such as solving x(x - 5) = 36 by first expanding and then moving all terms to one side.
• 😀 Learn how to solve quadratic equations by factoring, for instance, x^2 - 5x - 36 = 0 can be factored into (x - 9)(x + 4) = 0.
• 😀 Discover the process of verifying solutions by substituting the found values of x back into the original equation to confirm correctness.
• 😀 Understand how to expand and simplify quadratic expressions like (x+5)^2 + (x-2)^2 = 37 by applying algebraic expansion and combining like terms.
• 😀 Explore how to factor quadratic expressions such as 2x^2 + 6x - 8 into (2x - 2)(x + 4) = 0 and solve for x.
• 😀 Learn how to solve rational algebraic equations, like 6/x + (x-3)/4 = 2, by finding the LCD and eliminating denominators through multiplication.
• 😀 Recognize how to solve rational equations step by step, ensuring that the final result, like x = 3 or x = 8, satisfies the original equation by substitution.

Q & A

What is the first step in solving rational algebraic equations?

The first step is to find the least common denominator (LCD) of the terms involved in the equation.

How do you simplify the expression 1/x + 2x/5?

To simplify, first find the LCD, which is 5x. Then, multiply both sides by 5x to eliminate the denominators, simplifying the expression to x^2 + 5/5x.

In the equation x(x - 5) = 36, what form does the equation take after expanding and moving all terms to one side?

After expanding and moving all terms to one side, the equation becomes x^2 - 5x - 36 = 0, which is in the standard form of a quadratic equation.

How do you solve the quadratic equation x^2 - 5x - 36 = 0?

This equation can be factored into (x - 9)(x + 4) = 0. Solving for x gives x = 9 and x = -4.

When solving the equation x^2 + 10x + 25 + x^2 - 4x + 4 = 37, how do you combine like terms?

You combine like terms to get 2x^2 + 6x + 29 = 37. Then, subtract 37 from both sides to get the equation 2x^2 + 6x - 8 = 0.

How do you solve the equation 2x^2 + 6x - 8 = 0?

Factor the quadratic equation to get (2x - 2)(x + 4) = 0. Solving for x gives x = 1 and x = -4.

What is the method used to solve the rational algebraic equation 6/x + (x - 3)/4 = 2?

To solve, first find the LCD, which is 4x. Multiply both sides by 4x to eliminate the denominators, then simplify the resulting quadratic equation.

What happens when you multiply both sides of the equation 6/x + (x - 3)/4 = 2 by the LCD 4x?

Multiplying both sides by 4x cancels out the denominators, leading to the equation x^2 - 11x + 24 = 0, which is a quadratic equation.

How do you factor the quadratic equation x^2 - 11x + 24 = 0?

The equation factors as (x - 3)(x - 8) = 0. Solving for x gives x = 3 and x = 8.

What steps are involved in solving the equation x^4 - 5x^2 + 4 = 0?

First, rewrite the equation as a quadratic equation in terms of x^2: (x^2)^2 - 5x^2 + 4 = 0. Factor it as (x^2 - 4)(x^2 - 1) = 0. Then, solve for x by setting each factor equal to zero, giving x = 2, -2, 1, and -1.