Two distinct real roots: find them

Given that the quadratic equation x² − 6x + 5 = 0 has two distinct real roots, find its discriminant and the two roots.

Step-by-step

1. a = 1, b = -6, c = 5
   First read off the three coefficients, paying attention to the sign of the constant term.
2. Δ = b² − 4ac = -6² − 4 × 1 × 5 = 16
   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 = √16 = 4
   Take the square root of the discriminant; the quadratic formula uses its principal (non-negative) square root.
4. x = (−b ± √Δ)/(2a) ⇒ x_1 = 5, x_2 = 1
   The quadratic formula gives two values from the plus-minus sign, corresponding to the two roots.
5. x_1 + x_2 = −b/a = 6, x_1 · x_2 = c/a = 5
   Vieta's formulas can be used to verify whether the roots are correct.

Why it works

The sign of the discriminant determines the number of real roots: positive means two distinct real roots, zero means two equal real roots, and negative means no real roots. The problem stating there are two distinct real roots is telling you that the discriminant is positive, which serves as the basis for setting up an inequality or finding a parameter range. The discriminant only gives the number of roots; to find their specific values, you still need the quadratic formula.

Common mistakes

• 1) Treating "has two distinct real roots" and "has two real roots" as the same condition.
• 2) Calculating the discriminant correctly but forgetting to answer for the two roots asked simultaneously in the problem.
• 3) Confusing a positive discriminant with positive roots; there is no direct relationship between them.

Related

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