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
http://www.jiskha.com/display.cgi?id=1477323001
To determine the nature of the roots of a quadratic equation, we need to examine the discriminant. The discriminant is a value calculated from the coefficients of the quadratic equation. It helps us determine whether the equation has real or complex roots.
The discriminant (denoted as Δ) is calculated using the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
Now, let's determine the nature of the roots for each of the given quadratic equations:
(a) 3x^2 - 6x + 7 = 0
In this equation, a = 3, b = -6, and c = 7. Using the formula for the discriminant, Δ = (-6)^2 - 4 * 3 * 7 = 36 - 84 = -48.
Since the discriminant is negative (Δ < 0), this quadratic equation has two complex roots.
(b) 3x^2 + 6x - 7 = 0
In this equation, a = 3, b = 6, and c = -7. Using the formula for the discriminant, Δ = (6)^2 - 4 * 3 * (-7) = 36 + 84 = 120.
Since the discriminant is positive (Δ > 0), this quadratic equation has two distinct real roots.
(c) -5x^2 - x/5 = 0
To determine the nature of the roots, let's rewrite the equation in standard quadratic form. Simplifying, we get -5x^2 - x/5 = 0 is equivalent to -25x^2 - x = 0.
Now, in this equation, a = -25, b = -1, and c = 0. Using the formula for the discriminant, Δ = (-1)^2 - 4 * (-25) * 0 = 1 + 0 = 1.
Since the discriminant is positive (Δ > 0), this quadratic equation has two distinct real roots.
(d) 4x^2 - 28/3x + 49/9 = 0
To determine the nature of the roots, let's rewrite the equation in standard quadratic form. Simplifying, we get 4x^2 - (28/3)x + (49/9) = 0.
Now, in this equation, a = 4, b = -28/3, and c = 49/9. Using the formula for the discriminant, Δ = (-28/3)^2 - 4 * 4 * (49/9) = 784/9 - 784/9 = 0.
Since the discriminant is zero (Δ = 0), this quadratic equation has two identical real roots (a perfect square).
Therefore, the nature of the roots for each equation is:
(a) Two complex roots
(b) Two distinct real roots
(c) Two distinct real roots
(d) Two identical real roots (a perfect square)