Understanding Quadratics and Factorization

1. What is a Quadratic?

A quadratic equation is a second-degree polynomial, typically written in the form:

$ax^2 + bx + c = 0$

Here:

• $a$, $b$, and $c$ are constants (with $a \neq 0$).
• $x$ represents the variable.

Examples:

• $x^2 - 5x + 6 = 0$
• $2x^2 + 7x - 4 = 0$

2. What Does Factorizing Mean?

Factorizing a quadratic involves rewriting it as a product of two linear factors: $ax^2 + bx + c = (px + q)(rx + s)$

Where:

• $p$, $q$, $r$, and $s$ are constants.
• Expanding the product gives the original quadratic.

3. Steps to Factorize Quadratics

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Case 1: Simple Quadratics $(a = 1)$

If $a = 1$, the quadratic is: $x^2 + bx + c$

Steps:

1. Find two numbers that:

   • Multiply to $c$ (the constant term).
   • Add to $b$ (the coefficient of $x$).

2. Write the factors as: $x^2 + bx + c = (x + m)(x + n)$

Example:

Factorize $x^2 - 5x + 6$:

1. $c = 6$, $b = -5$.
2. Numbers are $-2$ and $-3$ because:
   • $(-2) \times (-3) = 6$
   • $(-2) + (-3) = -5$.
3. Write as: $x^2 - 5x + 6 = (x - 2)(x - 3)$

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Case 2: Quadratics with $a \neq 1$

For quadratics like $ax^2 + bx + c$, $a > 1$:

Steps:

1. Multiply $a \times c$.
2. Find two numbers that:
   • Multiply to $a \cdot c$.
   • Add to $b$.
3. Rewrite the middle term $bx$ using these numbers.
4. Group terms and factorize in pairs.
5. Write the result as a product of two binomials.

Example:

Factorize $2x^2 + 7x + 3$:

1. $a \cdot c = 2 \times 3 = 6$, $b = 7$.
2. Numbers are $6$ and $1$:
   • $6 \cdot 1 = 6$, $6 + 1 = 7$.
3. Rewrite $7x$ as $6x + x$: $2x^2 + 6x + x + 3$
4. Group terms: $(2x^2 + 6x) + (x + 3)$
5. Factorize each group: $2x(x + 3) + 1(x + 3)$
6. Factor out $(x + 3)$: $(2x + 1)(x + 3)$

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Case 3: Special Cases

• Perfect Squares: $x^2 + 6x + 9 = (x + 3)^2$
• Difference of Squares: $x^2 - 16 = (x - 4)(x + 4)$

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Practice Problems:

Try these for yourself:

1. $x^2 + 8x + 15$
2. $3x^2 + 11x + 10$
3. $x^2 - 9$
4. $2x^2 - 4x - 6$
5. $x^2 + 5x + 6$

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Tip:

Always check your factors by expanding them! This ensures they simplify back to the original quadratic.