Solving Quadratic Equations with the Quadratic Formula

The Quadratic Formula

For any quadratic equation written in the standard form:
$a{x}^2+bx+c=0$
where $a$, $b$, and $c$ are constants and $a\ne 0$, the solutions (roots) for $x$ are given by the quadratic formula:

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

This formula is one of the most important in algebra, and it's highly recommended to commit it to memory. The $\pm$ symbol (plus-minus) indicates that there are generally two solutions, one calculated with a plus sign and the other with a minus sign.

The process is essentially a three-step procedure:

1. Ensure the equation is in the standard form $a{x}^2+bx+c=0$.
2. Identify the values of $a$, $b$, and $c$.
3. Substitute these values into the formula and calculate the result.

Let's watch a video that introduces the formula and walks through the first example.

Dealing with Negative Coefficients

The most common source of errors when using the quadratic formula is mishandling negative signs. It is crucial to be methodical.

Let's focus on two specific scenarios:

1. When $c$ is negative: The term -4ac will become positive because of the double negative. For instance, if $c=-13$, then $-4ac=-4(a)(-13)$.
2. When $b$ is negative: This is particularly important.
   • The term $-b$ at the start of the formula will become positive. If $b=-11$, then $-b=-(-11)=11$.
   • The term $b^2$ must be calculated carefully. When you square a negative number, the result is positive. For instance, $(-11{)}^2=121$. When using a calculator, it is vital to use brackets, e.g., (-11)^2, to ensure the correct result. Typing -11^2 might be interpreted as -(11^2), giving you -121, which is incorrect.

The following segments of the video we just watched address these specific points.

To summarise the key points for handling negatives:

• Always write down your values for $a$, $b$, and $c$ before you start, including their signs.
• When substituting into the formula, use brackets for negative values, especially for the $b^2$ term: $(\text{negative }b{)}^2$.
• Be mindful of double negatives, such as in the $-b$ and $-4ac$ terms.

Exact Solutions (Surd Form)

Sometimes, particularly in non-calculator questions, you will be asked to give your answer in "exact form". This means leaving the square root in the answer, a form known as a surd. This requires you to simplify the surd if possible.

The process is the same, but instead of using a calculator at the end, you simplify the expression algebraically.

This method provides the precise mathematical answer, without any rounding. For a quick reference on the general method, the following resource is useful.

As we've seen before, the roots you calculate are the points where the parabola representing the function $y=a{x}^2+bx+c$ crosses the x-axis.

Conclusion

Congratulations! You have now mastered the most direct method for solving quadratic equations. The quadratic formula is your reliable tool for finding the roots of any quadratic, regardless of whether it can be factorised or how complex the numbers are.

Key Takeaways:

• The quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, solves any equation in the form $a{x}^2+bx+c=0$.
• The most crucial skill is substituting coefficients carefully, especially when they are negative. Use brackets!
• The $\pm$ sign yields two distinct solutions.
• Solutions can be expressed as decimal approximations (using a calculator) or as exact values in surd form (by simplifying algebraically).

For additional practice, you can work through the problems in 'Exercise 3' on page 7 of the "Quadratic Equations" PDF from mathcentre.ac.uk (LINK).

Preview of the Next Lesson:

We have now spent considerable time on quadratic polynomials (polynomials of degree 2), learning to factorise them and find their roots using various methods. In the next lesson, we will begin working with polynomials of higher degrees. We will start with a technique analogous to long division with numbers: polynomial long division. This is a fundamental skill that will allow us to factorise and find the roots of more complex polynomials like cubics and quartics.