3.2 Advanced Algebra

• In a quadratic equation, the constant \( a \) must not be equal to zero
• When factorizing \( 2x^2 + 7x + 6 = 0 \), the expression becomes \( (2x + 3)(x + 2) = 0 \)
• For the equation \( 2x^2 + 7x + 6 = 0 \), the values of \( a \), \( b \), and \( c \) are 2, 7, and 6
• What is factorisation in algebra?

  Expressing as a product of factors
• What is the factorisation of \( x^2 - 4 \) using the difference of squares technique?

  (x + 2)(x - 2)</latex>
• Order the steps in the factorisation process for simple quadratics:

  1️⃣ Identify the values of \( b \) and \( c \)

  2️⃣ Find two numbers \( p \) and \( q \) such that \( p + q = b \) and \( pq = c \)

  3️⃣ Write the expression as \( (x + p)(x + q) \)
• Factorisation simplifies algebraic expressions and aids in solving equations
• What is the process of expressing an algebraic expression as a product of two or more factors called?

  Factorization
• What is the result of factoring $a^{2} - b^{2}$ using the difference of squares technique?


  (a + b)(a - b)</latex>
• The expression $x^{2} - 4$ can be factored using the difference of squares method.


  True
• What is the first step in completing the square to solve a quadratic equation?

  Rearrange the equation
• Completing the Square is used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \), where \( a \) is not equal to zero
• Factorizing and using the quadratic formula are both primary methods to solve quadratic equations.


  True
• Setting each factor to zero after factorizing allows you to find the solutions of the equation.


  True
• The solutions for \( 2x^2 + 7x + 6 = 0 \) are \( x = -\frac{3}{2} \) and \( x = -2 \) using the quadratic formula.


  True
• In the common factor method, you identify a common term and factor it out
• In the simple quadratics technique, the values \( p \) and \( q \) must satisfy \( p + q = b \) and \( pq = c
• Factorizing is applicable to all quadratic equations.

  False
• What are the two possible values of $x$ when $a =$$2$, $b =$$7$, and $c =$$6$ in the quadratic formula?


  $- \frac{3}{2}$ and $- 2$
• Steps for factoring using the common factor technique

  1️⃣ Identify a common term in all terms

  2️⃣ Factor out the common term
• Factoring $x^{2} - 4$ using the difference of squares gives (x + 2)(x - 2)
• Steps to use the quadratic formula

  1️⃣ Identify $a$, $b$, and c</latex>

  2️⃣ Plug values into the formula

  3️⃣ Simplify under the square root

  4️⃣ Calculate the square root

  5️⃣ Find the two values of $x$
• When completing the square, the final step is to solve for $x$ by taking the square root of both sides.


  True
• Steps for completing the square in a quadratic equation

  1️⃣ Rearrange the equation to \( a(x^2 + bx) + c = 0 \)

  2️⃣ Add and subtract \( \left(\frac{b}{2a}\right)^2 \)

  3️⃣ Factor the left side to \( a(x + \frac{b}{2a})^2 + c - \left(\frac{b}{2a}\right)^2 = 0 \)

  4️⃣ Solve for \( x \) by taking the square root
• In the method of completing the square, you add and subtract the square of half the coefficient of \( x \), which is \(\left(\frac{b}{2a}\right)^2\)
• What value is added and subtracted in the equation \( 2(x^2 + 3x) - 5 = 0 \) to complete the square?

  $\frac{9}{4}$
• Match the method for solving quadratic equations with its description:

  Factorizing ↔️ Breaks down into binomial factors

  Quadratic Formula ↔️ Uses the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Match the method for solving \( 2x^2 + 7x + 6 = 0 \) with its description:

  Factorizing ↔️ Break down into binomial factors

  Quadratic Formula ↔️ Apply the formula with a = 2, b = 7, c = 6
• A quadratic equation can be written in the general form ax^2 + bx + c = 0
• Factorizing is simpler for equations that are easy to factor
• What is the first primary method to solve quadratic equations?

  Factorizing
• Factorisation simplifies algebraic expressions and aids in solving equations.


  True
• To factor a Simple Quadratic in the form x^2 + bx + c, you find two numbers p and q such that p + q = b and pq = c
• 4x + 8y can be factored as 4(x + 2y).


  True
• Factorization in algebra expresses an algebraic expression as a product of two or more factors.

  True
• What is the general form of a quadratic equation?

  $ax^{2} +$$bx +$$c =$$0$
• What does factorizing involve in the context of quadratic equations?

  Breaking down into binomial factors
• What is the quadratic formula?

  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}</latex>
• Match the method with its advantage or disadvantage:

  Factorizing ↔️ Simpler for easy-to-factor equations

  Quadratic Formula ↔️ Works for all quadratic equations