Which of the following equations has no real roots?
- $ x^2 + 4x + 3 \sqrt{2} = 0$
- $x^2 + 4x - 3 \sqrt{2} = 0$
- $x^2 + 5x + 3 \sqrt{2} = 0$
- $3x^2 + 4 \sqrt{3}x + 4 = 0$
Reveal answer
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A
Correct answer
Explanation
An equation has no real roots if the discriminant D < 0. For A: D = 4^2 - 4(1)(3*sqrt(2)) = 16 - 12*sqrt(2) = 16 - 16.97 < 0.
AI explanation
To find the equation with no real roots, we check the discriminant, D = b^2 - 4ac, for each equation; the equation must have a negative discriminant. For option A, x^2 + 4x + 3 sqrt(2) = 0, the values are a = 1, b = 4, and c = 3 sqrt(2). Calculating the discriminant gives D = 4^2 - 4(1)(3 sqrt(2)) = 16 - 12 sqrt(2). Since sqrt(2) is approximately 1.414, 12 sqrt(2) is about 16.97, making the discriminant 16 - 16.97 = -0.97. Because the discriminant is negative, the equation has no real roots.