Final answer:
The quadratic equation 2x^2 - 64x + 494 = 0 has two real roots, which are 19 and 13, determined by calculating the discriminant and applying the quadratic formula.
Step-by-step explanation:
To solve the quadratic equation 2x2 - 64x + 494 = 0, we can apply the quadratic formula, which is:
-b ± √(b² - 4ac) / 2a,
where a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0. For our equation:
- a = 2,
- b = -64, and
- c = 494.
First, we calculate the discriminant (Δ), which is b² - 4ac:
Δ = (-64)² - 4(2)(494).
If the discriminant is positive, the equation has two real roots. If it's zero, there is one real root. If the discriminant is negative, the equation has two complex roots. Let's calculate the discriminant:
Δ = 4096 - 4(2)(494) = 4096 - 3952 = 144.
Since Δ > 0, the equation has two real roots. We can now find these roots using the quadratic formula:
x = (-(-64) ± √144) / (2 * 2),
x = (64 ± 12) / 4,
giving two solutions:
- x = (64 + 12) / 4 = 76 / 4 = 19, and
- x = (64 - 12) / 4 = 52 / 4 = 13.
Therefore, the quadratic equation 2x2 - 64x + 494 = 0 has two real roots: 19 and 13.