Solving Quadratic Equations with the Quadratic Formula

The Quadratic Formula: Your Universal Tool

For any quadratic equation written in the standard form $a{x}^2+bx+c=0$, the solutions for $x$ are given by the formula:

The '±' symbol means that there are typically two solutions: one where we add the square root term, and one where we subtract it.

The most critical first step, before you do anything else, is to ensure your equation is in that standard form. This allows you to correctly identify the coefficients a, b, and c.

The image below shows the formula and two fully worked examples. Pay close attention to how the a, b, and c values are identified from the equation, including their signs, and then substituted into the formula.

The following video provides an excellent introduction to the formula and why the equation must equal zero.

Applying the Formula: A Step-by-Step Guide

Let's walk through an example. As per your preference, try to solve this equation first before reading the explanation:

Solve $x^2+5x+3=0$, giving your answer to 3 significant figures.

...

Here is the process:

1. Identify a, b, and c: The equation is already in standard form.

   • a = 1 (the coefficient of $x^2$)
   • b = 5 (the coefficient of $x$)
   • c = 3 (the constant term)

2. Substitute into the formula:

   $$x=\frac{-(5)\pm \sqrt{(5{)}^2-4(1)(3)}}{2(1)}$$

3. Simplify and Calculate:

   $$x=\frac{-5\pm \sqrt{25-12}}{2}$$ $$x=\frac{-5\pm \sqrt{13}}{2}$$

4. Find the two solutions: Now, we split the calculation for the '+' and '-' cases.

   • $x=\frac{-5+\sqrt{13}}{2}\approx \frac{-5+3.6055}{2}\approx -0.697$
   • $x=\frac{-5-\sqrt{13}}{2}\approx \frac{-5-3.6055}{2}\approx -4.30$

The video you just previewed walks through this exact problem. It's a great demonstration of how to use your calculator effectively.

Avoiding the Pitfalls: Common Mistakes

The quadratic formula is powerful, but it has several points where small mistakes, especially with signs, can lead to the wrong answer. This is where your focus on avoiding arithmetic errors will pay off.

The most common errors are:

1. Not rearranging the equation into standard form (ax^2 + bx + c = 0) first.
2. Sign errors with -b, especially when b itself is negative (e.g., if b = -5, then -b = -(-5) = 5).
3. Errors squaring a negative b value. Remember that (-b)^2 is always positive (e.g., (-5)^2 = 25, not -25). Using brackets in your calculator is essential here.
4. Sign errors with the -4ac term, especially when a or c is negative.

The following resource is structured perfectly for you. It presents four problems with flawed solutions. First, try to solve the equations yourself. Then, analyze the provided solutions to find the errors. This will actively train you to spot these common mistakes.

To further solidify your understanding of handling negative coefficients, which is often the trickiest part, the following video segment is extremely helpful.

Non-Calculator Questions and Surd Form

On the IGCSE non-calculator paper, you may be asked to give your answer in "exact form" or as a "simplified surd". This means you won't use a calculator to find a decimal approximation. Instead, you'll simplify the expression algebraically as much as possible.

The process involves two main stages of simplification:

1. Calculate the value under the square root (b^2 - 4ac), which is called the discriminant.
2. Simplify the resulting surd (e.g., sqrt(48) becomes 4*sqrt(3)).
3. Simplify the entire fraction, if possible.

The following guide and video will show you exactly how this is done.

The video below demonstrates this non-calculator process from start to finish.

Conclusion

You have now learned the most powerful and reliable method for solving quadratic equations. The quadratic formula provides a direct path to the solution for any quadratic, but its power comes with the responsibility of careful and accurate substitution.

Key Takeaways:

• The quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, solves any equation in the form $a{x}^2+bx+c=0$.
• Accuracy is paramount. Always start by writing the equation in standard form to identify a, b, and c correctly.
• Be extremely careful with negative signs, especially for the -b term and when squaring a negative b. Use parentheses when substituting into your calculator.
• For non-calculator papers, you must be able to simplify surds to give an exact answer.

In our next lesson, we will begin to combine our algebraic skills. We will tackle systems of simultaneous equations where one equation is linear and the other is quadratic. You will find that solving these systems often requires using the quadratic formula or one of the other methods we've learned in this module.