Solving Quadratic Equations by Factorising (equations with an x squared) (Junior Cert Mathematics): Revision Notes

Example Problem:

Solve the quadratic equation: $6x^2 + 11x - 10 = 0$

We'll solve this equation step by step using the Guide Number Method.

Step 1: Find the Guide Number

What we do:

Multiply the number in front of $x^2$ (here it's 6) by the last number (here it's -10).

Why we do it:

This multiplication gives us a new number, called the Guide Number. We'll use this Guide Number to help break the middle term into two smaller numbers.

$\text{Guide Number} = 6 \times (-10) = :highlight[-60]$

So, the Guide Number is -60.

Step 2: List All Factors of the Guide Number

What we do:

We list out all the pairs of numbers that multiply to give us the Guide Number (-60). These pairs are called factors.

Why we do it:

We need to find two numbers that not only multiply to give -60 but also add up to the middle number in the equation, which is 11. Listing all factors helps ensure we don't miss the right pair.

Factors of -60 are: $(-1, 60)$ $(1, -60)$ $(-2, 30)$ $(2, -30)$ $(-3, 20)$ $(3, -20)$ $(-4, 15)$ $(4, -15)$ $(-5, 12)$ $(5, -12)$ $(-6, 10)$ $(6, -10)$

Step 3: Identify Which Factors Add Up to the Middle Number

What we do:

Now, we look through the list of factors to find the pair that adds up to 11 (the number in front of $x$.

Why we do it:

The correct pair will help us split the middle term into two simpler terms, making the equation easier to factorise.

Looking at our factors, the correct pair is: $15 \times (-4) = -60 \quad \text{and} \quad 15 + (-4) = 11$

So, the numbers we need are 15 and -4.

Step 4: Rewrite the Middle Term Using These Factors

What we do:

We replace the middle term 11x in the original equation with the two numbers we found (15 and -4). This gives us:

$6x^2 + 15x - 4x - 10 = 0$

Why we do it:

By breaking the middle term into two parts, we make it possible to group and factor the equation in the next step.

Step 5: Factor by Grouping

What we do:

We now group the terms into two pairs and factor each pair separately.

$(6x^2 + 15x) - (4x + 10) = 0$

For the first group $6x^2 + 15x$, we can take out a common factor, which is 3x: $3x(2x + 5)$

For the second group $-4x - 10$, we can take out -2: $-2(2x + 5)$

So, now the equation looks like this: $3x(2x + 5) - 2(2x + 5) = 0$

Finally, because both parts have a common factor (2x + 5), we factor that out: $(3x - 2)(2x + 5) = 0$

Why we do it:

Grouping and factoring step by step helps us break the equation into two binomials (simpler expressions). Once we have these binomials, solving for $x$ is easy.

Step 6: Solve for $x$

What we do:

Set each binomial equal to zero and solve for $x$:

1. $3x - 2 = 0$ $3x = 2 \quad \Rightarrow \quad x = \frac{2}{3}$
2. $2x + 5 = 0$ $2x = -5 \quad \Rightarrow \quad x = -\frac{5}{2}$ So, the solutions are: $x = :success[\frac{2}{3}] \quad \text{or} \quad x = :success[-\frac{5}{2}]$

Why we do it:

Solving each binomial for $x$ gives us the possible values for $x$ that make the original equation true. These are the solutions to the quadratic equation.

Exam Tip

When solving quadratic equations, remember that you should always expect to find two solutions for $x$. This is because the squared term in the equation means that there are usually two values that can make the equation true. Always check that you've found both solutions before finishing the problem.