Factoring Quadratic Equations

Factoring Quadratic Equations

Factoring quadratic equations is a method to express a quadratic expression in the form $$ax^2 + bx + c$$ as a product of its linear factors $$(x - r_1)(x - r_2)$$, where $$r_1$$ and $$r_2$$ are the roots of the equation.

Steps for Factoring Quadratic Equations:

1. Identify the coefficients $$a$$, $$b$$, and $$c$$ in the quadratic equation $$ax^2 + bx + c = 0$$.
2. Find two numbers $$p$$ and $$q$$ such that:
   • $$p + q = b$$
   • $$pq = ac$$
3. Rewrite the middle term using $$p$$ and $$q$$: $$ax^2 + px + qx + c$$
4. Group the terms and factor out common factors: $$ax^2 + px + qx + c = (ax^2 + px) + (qx + c) = x(ax + p) + 1(qx + c)$$
5. Factor out the greatest common factor: $$(x(ax + p) + 1(qx + c)) = (x + \frac{c}{q})(ax + p)$$

Example:

Factor the quadratic equation: $$x^2 + 7x + 12 = 0$$

Solution:

1. Identify $$a=1$$, $$b=7$$, and $$c=12$$
2. Find $$p$$ and $$q$$: $$3 + 4 = 7$$ and $$3 \times 4 = 12$$
3. Rewrite: $$x^2 + 3x + 4x + 12$$
4. Group and factor: $$(x^2 + 3x) + (4x + 12) = x(x + 3) + 4(x + 3)$$
5. Factor out $$(x + 3)$$: $$(x + 3)(x + 4)$$

Therefore, the factored form is $$(x + 3)(x + 4) = 0$$

The roots of the equation are $$x = -3$$ and $$x = -4$$.

Quiz 1

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