Solve x² − 5x + 6: roots and Δ

Given a quadratic equation x² − 5x + 6 = 0 with two roots, find its discriminant and the two roots.

Step-by-step

1. a = 1, b = -5, c = 6
   First read off the three coefficients, paying attention to the sign of the constant term.
2. Δ = b² − 4ac = -5² − 4 × 1 × 6 = 1
   The discriminant determines the number of real roots: greater than zero gives two distinct real roots, equal to zero gives two equal real roots, less than zero gives no real roots.
3. √Δ = √1 = 1
   Take the square root of the discriminant; the quadratic formula uses its principal (non-negative) square root.
4. x = (−b ± √Δ)/(2a) ⇒ x_1 = 3, x_2 = 2
   The quadratic formula gives two values from the plus-minus sign, corresponding to the two roots.
5. x_1 + x_2 = −b/a = 5, x_1 · x_2 = c/a = 6
   Vieta's formulas can be used to verify whether the roots are correct.

Why it works

The core of solving a quadratic equation is the discriminant: it first tells you whether there are roots and how many, then determines whether to substitute into the quadratic formula. A positive discriminant means the graph intersects the x-axis at two points, so the plus and minus signs in the quadratic formula each correspond to one real root. After calculating, use the sum and product of the roots to check back; this immediately reveals any sign errors.

Common mistakes

• Writing the discriminant as b² + 4ac instead of b² − 4ac; one sign error makes the entire problem wrong.
• Writing the denominator in the quadratic formula as a instead of 2a.
• Assuming there is only one root when the discriminant is positive, and missing the other root.

Related

n-math-quadratic-none n-math-quadratic-roots-2 n-math-quadratic-discriminant-param