Quadratic Equation Solver

Step-by-step solution

1. 1

   Identify the coefficients

   a = 1, b = −5, c = 6.

   $$\begin{aligned}x^{2} - 5x + 6 &= 0 \\ a &= 1,\quad b = -5,\quad c = 6\end{aligned}$$
2. 2

   Compute the discriminant

   The discriminant is D = b² − 4ac.

   $$\Delta = b^{2} - 4ac = \left(-5\right)^{2} - 4 \cdot 1 \cdot 6 = 1$$

   • b² = (−5)² = 25
   • 4ac = 4 · 1 · 6 = 24
   • D = 25 − 24 = 1
3. 3

   Quadratic formula

   x = (−b ± √(b² − 4ac)) / (2a) = (−b ± √D) / (2a).

   $$\begin{aligned}x &= \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \\ &= \frac{5 \pm \sqrt{1}}{2}\end{aligned}$$
4. 4

   Interpret the discriminant

   D > 0: two distinct real roots.

5. 5

   Substitute the values

   x = (−(−5) ± √(1)) / (2).

   $$x = \frac{5 \pm \sqrt{1}}{2}$$

   • −b = 5
   • 2a = 2
6. 6

   Simplify the roots

   √D = √(1) = 1 is rational, giving two rational roots.

   $$\begin{aligned}x_{1} &= \frac{5}{2} + \frac{1}{2} = 3 \\ x_{2} &= \frac{5}{2} - \frac{1}{2} = 2\end{aligned}$$

   • x₁ = 5/2 + 1/2 = 3
   • x₂ = 5/2 − 1/2 = 2
7. 7

   Factored form

   (x − 3)(x − 2)

   $$\left(x - 3\right)\left(x - 2\right)$$
8. 8

   Vertex of the parabola

   x = −b / (2a) = 5/2, y = c − b² / (4a) = −1/4.

   $$\left(\frac{-b}{2a},\; c - \frac{b^{2}}{4a}\right) = \left(\frac{5}{2},\; \frac{-1}{4}\right)$$
9. 9

   Solution

   Roots: 3, 2.