Solve x² + 2x − 8: roots and Δ

Given a quadratic equation x² + 2x − 8 = 0 with two roots, find its discriminant and the two roots.

Step-by-step

1. a = 1, b = 2, c = -8
   First read off the three coefficients, paying attention to the sign of the constant term.
2. Δ = b² − 4ac = 2² − 4 × 1 × -8 = 36
   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 = √36 = 6
   Take the square root of the discriminant; the quadratic formula uses its principal (non-negative) square root.
4. x = (−b ± √Δ)/(2a) ⇒ x_1 = 2, x_2 = -4
   The quadratic formula gives two values from the plus-minus sign, corresponding to the two roots.
5. x_1 + x_2 = −b/a = -2, x_1 · x_2 = c/a = -8
   Vieta's formulas can be used to verify whether the roots are correct.

Why it works

The constant term in this problem is negative, so the −4ac part of the discriminant becomes adding a positive number, making the discriminant necessarily positive. The discriminant only determines the number and nature of the roots, not their magnitude; the magnitudes are given by the quadratic formula. When the constant term is negative, the two roots have opposite signs (one positive, one negative), which can be used to self-check whether the results are reasonable.

Common mistakes

• 1) Mixing up the negative sign in −4ac with the negative constant term, resulting in subtraction instead of addition.
• 2) Writing the square root of 36 as 36 when taking the square root.
• 3) Getting both roots as positive numbers without checking that a negative constant term implies opposite signs for the two roots.

Related

n-math-quadratic-roots-1 n-math-quadratic-discriminant-param n-math-quadratic-none