Use the discriminant to determine the nature of the roots.
1. 2x^2+5x-3=0
2. -x^2+x+1=0
b^2-4ac
5^2 -4(2)(-3)
that will be positive and therefore its square root is a real number. Thus two real roots, nothing imaginary about it
1^2 - 4(-1)(1)
everything positive again, two real roots
To determine the nature of the roots of a quadratic equation, we can use the discriminant, denoted as Δ (Delta). The discriminant is calculated by using the formula:
Δ = b^2 - 4ac
Where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
1. For the equation 2x^2 + 5x - 3 = 0:
Here, 'a' is 2, 'b' is 5, and 'c' is -3.
Therefore, we can calculate the discriminant as follows:
Δ = (5)^2 - 4(2)(-3)
Δ = 25 + 24
Δ = 49
Since the discriminant (Δ) is positive (greater than 0), the equation has two distinct real roots.
2. For the equation -x^2 + x + 1 = 0:
Here, 'a' is -1, 'b' is 1, and 'c' is 1.
Therefore, we can calculate the discriminant as follows:
Δ = (1)^2 - 4(-1)(1)
Δ = 1 + 4
Δ = 5
Since the discriminant (Δ) is positive (greater than 0), the equation has two distinct real roots.
In both cases, when the discriminant is greater than 0, it indicates that the quadratic equation has two real and distinct roots.