Lecture · 16 min · intermediate

Quadratics

Completing the square is the idea; the formula is the souvenir

Hall & Knight, Higher Algebra · quadratic equations

The quadratic formula is not a spell. It is completing the square, done once and for all, with letters instead of numbers. If you can complete the square, you never have to memorize the formula — and you can see the vertex of the parabola for free.

On the board today

—Complete the square for $x^{2} + b x + c$ .
—Derive the quadratic formula from $a x^{2} + b x + c = 0$ .
—Read vertex, axis, and number of real roots from the discriminant.

After completing the square

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

The discriminant $Δ = b^{2} - 4 a c$ counts real roots.

To complete the square of $x^{2} + b x$ , add and subtract $(b /2)^{2}$ . Then $x^{2} + b x + (b /2)^{2} = (x + b /2)^{2}$ . For $a x^{2} + b x + c$ , factor $a$ out of the $x$ -terms first. The vertex of $y = a (x - h)^{2} + k$ sits at $(h, k)$ : $h$ is the axis of symmetry, $k$ the extreme value.

Worked example · Complete the square

Rewrite $x^{2} - 6 x + 5$ as $a (x - h)^{2} + k$ , then solve $x^{2} - 6 x + 5 = 0$ .

1. $x^{2} - 6 x + 9 - 9 + 5 = (x - 3)^{2} - 4$ .
2. $(x - 3)^{2} = 4$ , so $x - 3 = \pm 2$ .
3. $x = 5$ or $x = 1$ .

Vertex at $(3, - 4)$ ; roots $1$ and $5$ .

Pause the lecture

If $Δ = b^{2} - 4 a c < 0$ , how many real roots? How many complex roots?

Common mistake. The $\pm$ is not optional decoration. Dropping it loses a root. And $2 a$ is the entire denominator — do not divide only $b$ by $2 a$ .

Practice this lecture

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