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Understanding Factoring and the Quadratic Formula in Algebra (Grade 9)

In algebra, factoring means rewriting an expression as a product (multiplication) of simpler expressions called factors. This skill helps simplify expressions and solve equations. There are several methods you can use:

1. Factoring Out the Greatest Common Factor (GCF)

• What It Is: Find a number, variable, or combination that is common to every term in the expression.
• Example:
    For 6x + 9, notice that both 6 and 9 can be divided by 3.
    Thus, factor out 3:
     6x + 9 = 3(2x + 3)

2. Factoring Quadratic Trinomials

• What It Is: These are expressions in the form ax² + bx + c. The goal is to rewrite them as the product of two binomials.
• Example:
    For x² + 5x + 6, find two numbers that multiply to 6 (the constant term) and add to 5 (the coefficient of x).
    Since 2 and 3 work (because 2 × 3 = 6 and 2 + 3 = 5), you can write:
     x² + 5x + 6 = (x + 2)(x + 3)

3. Factoring the Difference of Squares

• What It Is: When you have an expression of the form a² – b², it can be factored into two binomials: (a – b)(a + b).
• Example:
    For x² – 16, recognize that x² is the square of x and 16 is the square of 4.
    Thus, apply the formula:
     x² – 16 = (x – 4)(x + 4)

4. The Quadratic Formula

Sometimes a quadratic trinomial does not factor easily. In those cases, you can solve the quadratic equation using the quadratic formula.

• What It Is: For any quadratic equation in the form
     ax² + bx + c = 0
  the solutions for x can be found by using:
     x = (-b ± √(b² – 4ac)) / (2a)
• Example:
    Solve the equation 2x² + 7x + 3 = 0.
    Here, a = 2, b = 7, and c = 3.
    Plug these values into the formula:
     x = (–7 ± √(7² – 4·2·3)) / (2·2)
     x = (–7 ± √(49 – 24)) / 4
     x = (–7 ± √25) / 4
     x = (–7 ± 5) / 4
    Thus, the solutions are:
     x = (–7 + 5)/4 = –1/2
     x = (–7 – 5)/4 = –3

Why Are These Techniques Important?

• Simplification: Factoring helps simplify expressions, making them easier to work with.
• Solving Equations: Whether you factor or use the quadratic formula, these techniques allow you to find the values of x that satisfy an equation.
• Preparation: Mastering factoring and the quadratic formula sets a strong foundation for more advanced algebra topics.

Practice these methods with different problems to build your understanding and confidence in algebra. Both factoring and the quadratic formula are essential tools for solving quadratic equations. Now try to solve the worksheet provided with grades 9 and 10 Math teacher

“In Grade 9 Ontario Math, students are introduced to foundational concepts that prepare them for higher-level math courses. This worksheet, provided by Khoda Zamani, an OCT-certified teacher, is designed to help students review key topics for their final exam. Khoda Zamani also offers math tutoring and physics tutoring to ensure students have the support they need to succeed.”

Grade 11 Physics Review Worksheet! This resource has been designed by a  ontario certified Physics teacher  to help you consolidate and practice key concepts covered in the Ontario curriculum and the Nelson Physics textbook. It covers topics like kinematics, dynamics, energy, waves, and more, providing detailed solutions to enhance your understanding. Use this worksheet as a preparation tool for quizzes, tests, and exams to build confidence and excel in Grade 11 Physics.

Sound needs a medium to travel. The speed of sound needs a medium to carry on their wave.

When the temperature of the medium increases the molecules have more energy as a result molecules move faster so the speed of sound can travel faster.

The speed of sound in air at 0 ˚c is 346 m/s.

so

V=346°c

If air temperature increases by 1°c the speed increase by 0.606

So our formula will be

V=331.4m/s+(0.606m/s/˚c)T

Which T indicates the temperature of its units in Celsius.

Example:

Determine the speed of sound at 45.0˚c

Given:T=45.0˚c

Formula:

V=331.4m/s+(0.606m/s/˚c)T

V=331.4m/s+(0.606m/s/˚c)45      (substitude the T in formula)

V=358.67 m/s