The Quadratic Formula
Derive the quadratic formula by completing the square on ax² + bx + c = 0. Apply x = (−b ± √(b²−4ac))/2a to solve any quadratic. Understand it as the master tool that subsumes factoring and square-root methods.
A lesson of Algebra 1, in Quadratic Functions, about 35 minutes.
By the end of this lesson, you will be able to identify $a$, $b$ and $c$ correctly, apply the formula without sign errors, simplify the result, and choose sensibly between the formula and factoring.
One Formula That Always Works
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
For any equation in the form $ax^2 + bx + c = 0$. Not most of them. Any of them.
Where it comes from is the previous lesson: completing the square on $ax^2 + bx + c = 0$ with letters instead of numbers produces exactly this. It is not a new idea, it is a computation done once so nobody has to repeat it.
Standard form first, always. The formula reads $a$, $b$ and $c$ off an equation that says $= 0$. Applying it to $x^2 + 3x = 10$ with $c = 10$ gives a wrong answer with no warning, and it is the single most common way the formula fails.
$$x^2 + 3x = 10 \quad \Rightarrow \quad x^2 + 3x - 10 = 0 \quad \Rightarrow \quad c = -10$$
Reading a, b and c
The signs belong to the coefficients. Take them across.
| Equation | $a$ | $b$ | $c$ |
|---|---|---|---|
| $x^2 + 5x + 6 = 0$ | $1$ | $5$ | $6$ |
| $2x^2 - 3x + 1 = 0$ | $2$ | $-3$ | $1$ |
| $x^2 - 4 = 0$ | $1$ | $0$ | $-4$ |
| $-x^2 + 2x + 8 = 0$ | $-1$ | $2$ | $8$ |
| $3x^2 = 7x$ | $3$ | $-7$ | $0$ |
A missing term means a coefficient of zero, not a missing coefficient. And the last row had to be rearranged to $3x^2 - 7x = 0$ first, which flipped the sign of the $7$.
An invisible $1$ is still a $1$. In $x^2 - 4 = 0$, $a$ is $1$.
Sort It: What Is $c$
Rearrange to standard form in your head first.
Rows one and two are the same equation and the same answer, which is the point: the rearranging is where the sign changes, and doing it first makes the rest mechanical.
Example 1: The Whole Method
Here is every step on one equation, worked all the way through. $$2x^2 + 5x - 3 = 0$$
- What is a? 2
- What is b? 5
- What is c? -3
- Work out b squared minus 4ac 49
- What is the square root of 49? 7
- The formula is (-5 plus or minus 7) over 4. What does the plus branch give? 1/2