Learn: Quadratic Equations

The idea

A quadratic is an equation whose biggest power is x². Its graph is a U-shaped curve called a parabola, and solving the equation finds where that curve crosses the x-axis. For x² − 5x + 6 = 0, ask: which two numbers multiply to +6 and add to −5?

Worked example

Every step explained — why we do it, and what happens.

1. 1
   $x^2 - 5x + 6 = 0$

   Why: Spot the quadratic: x² is the biggest power, so the graph is a parabola. The "= 0" means we are hunting where the curve touches the x-axis — those spots are the roots.

   What happens: Find the x-values where the parabola crosses the axis.

2. 2
   $(x - 2)(x - 3) = 0$

   Why: Factor: find two numbers that multiply to +6 AND add to −5. −2 and −3 do the job. Factoring turns one tricky question into two easy ones.

   What happens: The quadratic becomes a product of two brackets.

3. 3
   $x - 2 = 0 \text{ or } x - 3 = 0$

   Why: The zero-product rule: if two things multiply to 0, one of them must BE 0. (A product can only be 0 when a factor is 0.)

   What happens: One big equation splits into two tiny ones.

4. 4
   $x = 2 \text{ or } x = 3$

   x − 2 = 0 → x = 2; x − 3 = 0 → x = 3.

5. 5
   $2^2 - 5(2) + 6 = 0 \qquad \text{and} \qquad 3^2 - 5(3) + 6 = 0$

   Why: The check: plug each root back into the original to confirm the parabola really crosses there.

   What happens: 4 − 10 + 6 = 0 ✓ and 9 − 15 + 6 = 0 ✓ — both roots are true. The parabola crosses the x-axis twice.