Solving Quadratic Equations using the '-b Formula' (Junior Cert Mathematics): Revision Notes

Solving Quadratic Equations with a formula

What is the $-b$ Formula?

The $-b$ formula, also known as the Quadratic Formula, is a method used to solve any quadratic equation. Unlike factorising, which works only when a quadratic can be easily broken down into simpler factors, the $-b$ formula works for all quadratic equations.

A quadratic equation looks like this: $ax^2 + bx + c = 0$ where:

• $x^2$ is the squared term.
• $a$, $b$, and $c$ are numbers.
• $x$ is the variable we want to solve for. The formula to solve any quadratic equation is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Steps to Solve a Quadratic Equation Using the $-b$ Formula

Let's solve a quadratic equation step by step using the $-b$ formula.

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Step 3: Simplify Inside the Square Root

What we do:

First, calculate what's inside the square root: $x = \frac{8 \pm \sqrt{64 - 48}}{4}$

Let's break that down:

• The $-(-8)$ becomes $8$.
• The $64$ comes from $(-8)^2 = 64$.
• The $48$ comes from $4(2)(6) = 48$. So now, we have: $x = \frac{8 \pm \sqrt{16}}{4}$

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Step 4: Calculate the Square Root and Simplify Further

What we do:

Now, we find the square root: $\sqrt{16} = 4$

So the equation becomes: $x = \frac{8 \pm 4}{4}$

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Step 5: Solve for $x$

What we do:

Split the equation into two parts to find the two possible solutions for $x$:

1. $x = \frac{8 + 4}{4}$ $x = \frac{12}{4} = 3$
2. $x = \frac{8 - 4}{4}$ $x = \frac{4}{4} = 1$ So, the solutions are: $x = 3 \quad \text{or} \quad x = 1$

Why we do it:

Quadratic equations typically have two solutions. This final step gives us the exact values of $x$ that make the original equation true.

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Summary

Here's a quick recap of what we did:

1. Identify the Values: Find the values of $a$,$b$, and $c$ in the quadratic equation.
2. Plug into the Formula: Substitute these values into the $-b$ formula, using brackets to avoid mistakes.
3. Simplify: Calculate what's inside the square root, then find the square root.
4. Solve for $x$: Split the equation and solve for the two possible values of $x .$ By following these steps, you can solve any quadratic equation using the $-b$ formula. Remember, practice is key, and taking your time to understand each step will make this method much easier!

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