Determine the nature of the roots of the following quadratic equations:
(a) 3x^2 - 6x + 7 = 0
(b) 3x^2 + 6x - 7 = 0
(c) -5x^2- x/5 = 0
(d) 4x^2 - 28/3x + 49/9 = 0
check the discriminants
To determine the nature of the roots of a quadratic equation, we need to analyze the discriminant of the equation. The discriminant is the part of the quadratic formula inside the square root sign, given by b^2 - 4ac, where the quadratic equation is in the form ax^2 + bx + c = 0.
(a) For 3x^2 - 6x + 7 = 0:
Here, a = 3, b = -6, and c = 7.
The discriminant is calculated as b^2 - 4ac = (-6)^2 - 4(3)(7) = 36 - 84 = -48.
Since the discriminant is negative (-48 < 0), the equation has complex roots, meaning that it does not have real roots.
(b) For 3x^2 + 6x - 7 = 0:
Here, a = 3, b = 6, and c = -7.
The discriminant is calculated as b^2 - 4ac = (6)^2 - 4(3)(-7) = 36 + 84 = 120.
Since the discriminant is positive (120 > 0), the equation has two distinct real roots.
(c) For -5x^2 - x/5 = 0:
First, let's simplify the equation by multiplying both sides by 5 to get rid of the fraction.
The equation becomes -25x^2 - x = 0.
Here, a = -25, b = -1, and c = 0.
The discriminant is calculated as b^2 - 4ac = (-1)^2 - 4(-25)(0) = 1.
Since the discriminant is positive (1 > 0), the equation has two distinct real roots.
(d) For 4x^2 - 28/3x + 49/9 = 0:
First, let's simplify the equation by multiplying both sides by 9 to get rid of the fractions.
The equation becomes 36x^2 - 84x + 49 = 0.
Here, a = 36, b = -84, and c = 49.
The discriminant is calculated as b^2 - 4ac = (-84)^2 - 4(36)(49) = 7056 - 7056 = 0.
Since the discriminant is zero (0 = 0), the equation has two equal real roots.