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

How we read your problem

A quadratic equation in x: solve with the quadratic formula.

Step-by-step solution

1. 1

   Start with the equation

   $${ x}^{2}-5~ x+6 = 0$$
2. 2

   Move every term to one side to get standard form

   $$x^{2} - 5x + 6 = 0$$
3. 3

   Compute the discriminant

   $$\Delta = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 6 = 1$$
4. 4

   Δ > 0, so there are two distinct real roots

5. 5

   Apply the quadratic formula

   $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$$
6. 6

   Simplify each root

   $$x_1 = \frac{5 - 1}{2} = 2,\quad x_2 = \frac{5 + 1}{2} = 3$$
7. 7

   Factored form

   $$x^2 - 5x + 6 = (x - 2)(x - 3)$$

Formulas used

• Quadratic formula

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

• Discriminant

  $$\Delta = b^2 - 4ac$$

• Vertex

  $$\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)$$

The quadratic formula gives every solution of ax² + bx + c = 0.

Verification

• x = 2: left side = 0, right side = 0 ✓
• x = 3: left side = 0, right side = 0 ✓