Quadratic EquationsThe Quadratic Formula

Completing the square can be tricky, and it is easy to make mistakes along the way.

Let's follow the steps when completing the square, but use a, b and c as coefficients for the quadratic equation, rather than actual numbers:

Completing the square is long and complicated, and it is easy to make mistakes. Luckily, there is a shortcut that makes it a lot simpler!

To find it, we need to repeat the process of completing the square, but leaving the coefficients as a, b and c rather than actual numbers.

Lets start with a quadratic equation of the form

ax2+bx+c=0.

To make the first term a perfect square, we have to multiply the entire equation by a. To avoid fractions, we also multiply by 4. This gives;

4a2x2+4abx+4ac=0.

Now apply the box method:

| | 2ax | b | | 2ax | 4a2x2 | 2abx | | b | 2abx | b2 | {.q-grid}

These steps were ugly, painful, and you don't need to remember them (even though it was just the same as completing the square, just with variables). The result, however, was worth it: a single equation that tells us the solutions of any quadratic equation. It is often called the Quadratic Formula:

x=−b±b2−4ac2a

To solve a quadratic equation, we just have to replace a, b and c with the actual numbers in our case, and then simplify the fraction.

Some curricula feel it is important to notice that the formula x=−b±b2−4ac√2a represents two symmetrical values about the middle point x=−b2a.

From part 4 of this course we know that the vertex of the parabola lies halfway between any two symmetric points. Our technique of simply looking for interesting x-values makes the location of the vertex clear. One need not know this formula.

(But if you do want it … just write y=ax2+bx+c as y=x(ax+b)+c. This shows that inputs x=0 and x=−ba give symmetrical outputs. The vertex is thus halfway between these values … at x=−b2a!)

The Discriminant

One particularly important part of the quadratic equation is the term under the square root, which is called the discriminant. Depending on the value of b2−4ac, you can tell a lot about the solutions of a quadratic equation, without ever actually soling it:

• If b2−4ac<0, the quadratic equation has no solutions, because we cannot take square roots of negative numbers. (More on that later…)
• If b2−4ac=0, the quadratic equation has one solution. Zero is the only number with just one square root, because +0=−0.
• If b2−4ac>0, the quadratic equation has two solutions like before, one when evaluating the quadratic formula with +, and one when evaluating it with –.

Exercises

The two solutions lie at symmetrical positions about the value x=−b2a.

Solving Quadratic Equations – Summary

We now saw multiple different ways to solve quadratic equations, all of which have advantages and disadvantages:

• Basic Algebra This is the easiest way, but it only works for quadratic equations that don't contain an x-term.

• Factoring Also quite simply, but it takes some guesswork and it doesn't always work.

• Completing the Square Very long and complicated. It is easy to make mistakes. In addition to finding the solutions of an equation, it also tells us the vertex of the corresponding parabola.

• Quadratic Formula Straightforward formula that always work, but it sometimes feels like "magic" and it is easy to forget why and how it works.

Final Exercises