Use the discriminant to determine the number and type of solutions for this equation x^2-3x+5=0
To use the discriminant, we need to first identify the coefficients from the quadratic equation. In this case, the quadratic equation is:
ax^2 + bx + c = 0
Comparing this with the given equation, x^2 - 3x + 5 = 0, we can find that:
a = 1
b = -3
c = 5
The discriminant, represented as Δ, is calculated as:
Δ = b^2 - 4ac
Substituting the values from our equation, we can compute:
Δ = (-3)^2 - 4(1)(5)
= 9 - 20
= -11
Now, let's analyze the discriminant to determine the number and type of solutions based on its value:
1. If the discriminant (Δ) is positive (Δ > 0), then the quadratic equation has two distinct real solutions.
2. If the discriminant (Δ) is zero (Δ = 0), then the quadratic equation has one real solution (also known as a double root).
3. If the discriminant (Δ) is negative (Δ < 0), then the quadratic equation has no real solutions. However, there are two complex conjugate solutions.
In our case, the discriminant is -11 (Δ = -11), which is negative. Therefore, the equation x^2 - 3x + 5 = 0 has no real solutions but two complex conjugate solutions.
b^2 - 4ac
= 9 - 4(1)(5)
= 9-20
= -11
there will be two imaginary roots