Quadratic equations — factoring and the quadratic formula (KCSE Mathematics Form 3)

TL;DR

Quadratic equations are equations with an $x^2$ term, and you'll learn two main ways to solve them: factoring and using the quadratic formula. Factoring works when you can easily break down the expression into two brackets, while the quadratic formula is a universal tool for all quadratic equations. Both methods help you find the values of $x$ that make the equation true.

1. The Mental Model

Imagine you have a puzzle where you need to find specific numbers that fit into an equation involving a squared term. Factoring is like finding the right pieces to fit together, while the quadratic formula is a special key that unlocks the answer every time, even for tricky puzzles.

2. The Core Material

A quadratic equation is any equation that can be written in the standard form:
$ax^2 + bx + c = 0$
where $a$, $b$, and $c$ are numbers, and $a$ cannot be zero. If $a$ were zero, it wouldn't be quadratic anymore! The solutions to a quadratic equation are also called its roots.

Solving by Factoring

Factoring is a method that works well when the quadratic expression can be easily broken down into a product of two linear expressions (two brackets).

Steps for Factoring:
1. Ensure the equation is in standard form: $ax^2 + bx + c = 0$.
2. Find two numbers: These two numbers must multiply to give $ac$ and add up to $b$.
3. Rewrite the middle term: Replace $bx$ with the two terms you found in step 2.
4. Factor by grouping: Group the terms into pairs and factor out the common factor from each pair.
5. Set each factor to zero: Once you have $(px + q)(rx + s) = 0$, set $px + q = 0$ and $rx + s = 0$ and solve for $x$.

Example: Solve $x^2 + 5x + 6 = 0$ by factoring.
1. Already in standard form. Here, $a=1, b=5, c=6$.
2. We need two numbers that multiply to $ac = 1 \times 6 = 6$ and add to $b = 5$. These numbers are 2 and 3.
3. Rewrite: $x^2 + 2x + 3x + 6 = 0$.
4. Factor by grouping:
$x(x + 2) + 3(x + 2) = 0$
$(x + 2)(x + 3) = 0$
5. Set each factor to zero:
$x + 2 = 0 \implies x = -2$
$x + 3 = 0 \implies x = -3$
So, the roots are $x = -2$ and $x = -3$.

Solving using the Quadratic Formula

The quadratic formula is a powerful tool because it works for any quadratic equation, even those that are difficult or impossible to factor.

The formula is:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Steps for using the Quadratic Formula:
1. Ensure the equation is in standard form: $ax^2 + bx + c = 0$.
2. Identify $a$, $b$, and $c$: Carefully note the coefficients, including their signs.
3. Substitute the values into the formula: Be very careful with negative signs.
4. Simplify: Calculate the value under the square root (the discriminant, $b^2 - 4ac$) first, then simplify the square root, and finally perform the addition/subtraction and division.

Example: Solve $2x^2 - 5x - 3 = 0$ using the quadratic formula.
1. Already in standard form.
2. Identify $a=2, b=-5, c=-3$.
3. Substitute into the formula:
$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}$
4. Simplify:
$x = \frac{5 \pm \sqrt{25 - (-24)}}{4}$
$x = \frac{5 \pm \sqrt{25 + 24}}{4}$
$x = \frac{5 \pm \sqrt{49}}{4}$
$x = \frac{5 \pm 7}{4}$

This gives two solutions:
$x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3$
$x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$
So, the roots are $x = 3$ and $x = -\frac{1}{2}$.

When to use which method?

graph TD
    A[Start: Quadratic Equation ax² + bx + c = 0] --> B{Can it be easily factored?};
    B -- Yes --> C[Use Factoring Method];
    C --> D[Find two numbers that multiply to ac and add to b];
    D --> E[Rewrite middle term, factor by grouping];
    E --> F[Set each factor to zero and solve for x];
    B -- No --> G[Use Quadratic Formula];
    G --> H[Identify a, b, c];
    H --> I[Substitute into x = (-b ± sqrt(b² - 4ac)) / 2a];
    I --> J[Simplify to find x];
    F --> K[End: Solutions (roots) for x];
    J --> K;

3. Worked Example

Problem: Solve the equation $3x^2 + 10x = 8$.

Solution:

Step 1: Rewrite in standard form.
The given equation is $3x^2 + 10x = 8$.
To get it into $ax^2 + bx + c = 0$ form, subtract 8 from both sides:
$3x^2 + 10x - 8 = 0$

Step 2: Choose a method.
Let's try factoring first. We need two numbers that multiply to $ac = 3 \times (-8) = -24$ and add to $b = 10$.
After some thought, the numbers 12 and -2 fit these conditions ($12 \times -2 = -24$ and $12 + (-2) = 10$).

Step 3: Solve by Factoring.
Rewrite the middle term:
$3x^2 + 12x - 2x - 8 = 0$
Factor by grouping:
$3x(x + 4) - 2(x + 4) = 0$
$(3x - 2)(x + 4) = 0$
Set each factor to zero:
$3x - 2 = 0 \implies 3x = 2 \implies x = \frac{2}{3}$
$x + 4 = 0 \implies x = -4$

So, the solutions are $x = \frac{2}{3}$ and $x = -4$.

Step 4: (Optional, for verification) Solve using the Quadratic Formula.
From $3x^2 + 10x - 8 = 0$, we have $a=3, b=10, c=-8$.
Substitute into the formula:
$x = \frac{-10 \pm \sqrt{10^2 - 4(3)(-8)}}{2(3)}$
$x = \frac{-10 \pm \sqrt{100 - (-96)}}{6}$
$x = \frac{-10 \pm \sqrt{100 + 96}}{6}$
$x = \frac{-10 \pm \sqrt{196}}{6}$
$x = \frac{-10 \pm 14}{6}$

Two solutions:
$x_1 = \frac{-10 + 14}{6} = \frac{4}{6} = \frac{2}{3}$
$x_2 = \frac{-10 - 14}{6} = \frac{-24}{6} = -4$

Both methods give the same solutions, $x = \frac{2}{3}$ and $x = -4$.

4. Key Takeaways

• A quadratic equation has the highest power of $x$ as 2, and its standard form is $ax^2 + bx + c = 0$.
• Factoring involves breaking down the quadratic expression into a product of two linear factors.
• The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, always works to find the roots.
• Always ensure the equation is in standard form before applying either method.
• The term $b^2 - 4ac$ (the discriminant) tells you about the nature of the roots (e.g., if it's negative, there are no real solutions).
• Quadratic equations usually have two solutions (roots), but sometimes they can have one repeated solution or no real solutions.

Common Mistakes to Avoid:
- Forgetting to set the equation to $0$ before factoring or identifying $a, b, c$.
- Making sign errors, especially when substituting negative values into the quadratic formula.
- Incorrectly simplifying the square root or the entire quadratic formula expression.
- Trying to factor when the numbers are difficult, instead of switching to the quadratic formula.

5. Now Try It

Solve the equation $4x^2 - 7x = 2$ using both factoring and the quadratic formula. Show all your steps clearly for both methods.

What success looks like: You should arrive at the same two solutions for $x$ using both methods, and your working should be neat and easy to follow. You should be able to state the values of $a, b, c$ correctly for the quadratic formula, and find the correct pair of numbers for factoring.