Factoring quadratic equations is an important skill you'll learn in Year 10 Math. This skill helps you find the solutions, or roots, of quadratic equations. These equations usually look like this:

$$ax^2 + bx + c = 0$$

Here, $a$, $b$, and $c$ are numbers, and $a$ can't be zero. The goal of factoring is to rewrite the quadratic equation as a product of two smaller expressions, making it easier to solve for $x$. Let’s go through the basic steps of factoring quadratic equations together.

Step 1: Identify the Coefficients

Start by finding the values of $a$, $b$, and $c$. Knowing these numbers is very important because they help us with the factoring process.

Example: In the equation $2x^2 + 5x + 3 = 0$, we find:

• $a = 2$
• $b = 5$
• $c = 3$

Step 2: Multiply $a$ and $c$

Now, multiply $a$ and $c$ together:

$$ac = a \times c$$

This product will help us find two numbers that add up to $b$ and multiply to $ac$.

Example: For $2x^2 + 5x + 3 = 0$, we calculate $ac = 2 \times 3 = 6$.

Step 3: Find Two Numbers

Next, look for two whole numbers that:

• Multiply to $ac$
• Add up to $b$

Sometimes, you might need to try different combinations.

Example: We need two numbers that multiply to $6$ and add to $5$. The numbers $2$ and $3$ work because:

• $2 \times 3 = 6$
• $2 + 3 = 5$

Step 4: Rewrite the Middle Term

Now, use those two numbers to rewrite the quadratic equation. Replace the middle term ($bx$) with the two numbers you found.

Example: The equation $2x^2 + 5x + 3$ can be rewritten as:

$$2x^2 + 2x + 3x + 3 = 0$$

Step 5: Factor by Grouping

Group the expression into two pairs and factor each pair:

$$(2x^2 + 2x) + (3x + 3) = 0$$

Now, factor out the common factors from each pair:

$$2x(x + 1) + 3(x + 1) = 0$$

Step 6: Factor Out the Common Binomial

Next, take out the common binomial (which is $x + 1$) from the groups:

$$(2x + 3)(x + 1) = 0$$

Step 7: Set Each Factor to Zero

Now, set each factor equal to zero to find $x$:

1. $2x + 3 = 0$
2. $x + 1 = 0$

Step 8: Solve for $x$

Solving these equations gives us:

1. $2x + 3 = 0 \Rightarrow 2x = -3 \Rightarrow x = -\frac{3}{2}$
2. $x + 1 = 0 \Rightarrow x = -1$

Conclusion

So, the roots of the quadratic equation $2x^2 + 5x + 3 = 0$ are $x = -\frac{3}{2}$ and $x = -1$.

Factoring quadratic equations is a key algebra skill. Many students see a big improvement—about 60%—in their problem-solving skills by practicing this. Learning how to factor will help you as you continue with more advanced math topics, especially those related to polynomials and quadratic functions.