how do I discuss or describe the nature of the roots for this equation: 3x^2 - 5x = 2
That is a quadratic.
Do you know how to use the quadratic equation to solve
3 x^2 - 5 x - 2 = 0 ?
If not factor
(3x + 1)(x-2) = 0
that is true if
x = 2
or if
x = -1/3
For ax^2 + bx + c = 0
the sum of the roots = -b/a
the product of the roots = c/a
so for 3x^2 - 5x - 2 = 0
the sum of the roots = ....
the product of the roots = ....
To discuss or describe the nature of the roots for the equation 3x^2 - 5x = 2, you can follow these steps:
Step 1: Rewrite the equation in the standard form:
3x^2 - 5x - 2 = 0
Step 2: Determine the discrimi
To discuss or describe the nature of the roots of the equation 3x^2 - 5x = 2, we first need to solve the equation and find the values of x.
Step 1: Rewrite the equation in standard form:
3x^2 - 5x - 2 = 0
Step 2: We need to find the roots by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the roots can be calculated as follows:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation (3x^2 - 5x - 2 = 0), a = 3, b = -5, and c = -2. Substitute these values into the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4 * 3 * -2)) / (2 * 3)
Step 3: Simplify the equation using basic arithmetic:
x = (5 ± √(25 + 24)) / 6
x = (5 ± √(49)) / 6
x = (5 ± 7) / 6
Step 4: Determine the values of x:
x1 = (5 + 7) / 6 = 12/6 = 2
x2 = (5 - 7) / 6 = -2/6 = -1/3
So, the equation 3x^2 - 5x = 2 has two roots: x = 2 and x = -1/3.
To discuss the nature of these roots, we can analyze the discriminant of the quadratic equation. The discriminant is the expression inside the square root in the quadratic formula, (b^2 - 4ac).
If the discriminant is greater than 0, the equation has two distinct real roots.
If the discriminant is equal to 0, the equation has one real root (which is repeated).
If the discriminant is less than 0, the equation has two complex conjugate roots.
In our equation, the discriminant is (b^2 - 4ac) = (-5)^2 - 4 * 3 * -2 = 49. Since the discriminant is positive (greater than 0), the equation has two distinct real roots.