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 49
----x + ----- = 0
3 9
b^2-4ac says it all
in (a) for example
36-4(21) is negative so sqrt of negative in sollution so complex roots
(b) 36 + 4(21) is + so real roots
(c) x^2 + x/25 = 0
x(x+1/25) = 0
x = 0 or x = -1/25
(d) ?
check in your text or in your notes what properties are needed to investigate the nature of the roots.
hint: it has to do with the determinant.
btw. - I can't make your your last equation.
To determine the nature of the roots of a quadratic equation, we need to consider the discriminant (the expression inside the square root in the quadratic formula). The discriminant tells us whether the equation has real roots, imaginary roots, or repeated roots.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Now let's analyze each equation:
(a) 3x^2 - 6x + 7 = 0
In this equation, a = 3, b = -6, and c = 7.
The discriminant is: b^2 - 4ac = (-6)^2 - 4(3)(7) = 36 - 84 = -48
Since the discriminant (-48) is negative, the roots are complex, and the equation has two distinct complex roots.
(b) 3x^2 + 6x - 7 = 0
In this equation, a = 3, b = 6, and c = -7.
The discriminant is: b^2 - 4ac = (6)^2 - 4(3)(-7) = 36 + 84 = 120
Since the discriminant (120) is positive, the roots are real, and the equation has two distinct real roots.
(c) -5x^2 - x/5 = 0
This equation is not in the standard quadratic form (ax^2 + bx + c = 0) since there is a division by 5. Let's simplify it first:
-5x^2 - x/5 = 0
Multiply through by 5 to clear the fraction:
-25x^2 - x = 0
Now we can apply the quadratic formula. In this equation, a = -25, b = -1, and c = 0.
The discriminant is: b^2 - 4ac = (-1)^2 - 4(-25)(0) = 1
Since the discriminant (1) is positive, the roots are real, and the equation has two distinct real roots.
(d) 4x^2 - 28 49
----x + ---- = 0
3 9
To simplify this equation, we need to multiply through by the denominators to eliminate fractions:
4x^2 * 9 - 28x + 49x = 0
36x^2 - 28x + 49x = 0
36x^2 + 21x = 0
Now let's apply the quadratic formula. In this equation, a = 36, b = 21, and c = 0.
The discriminant is: b^2 - 4ac = (21)^2 - 4(36)(0) = 441
Since the discriminant (441) is positive, the roots are real, and the equation has two distinct real roots.
So, to summarize:
(a) The equation 3x^2 - 6x + 7 = 0 has two distinct complex roots.
(b) The equation 3x^2 + 6x - 7 = 0 has two distinct real roots.
(c) The equation -5x^2 - x/5 = 0 has two distinct real roots.
(d) The equation 4x^2 + 21x = 0 has two distinct real roots.