To find the roots of the quadratic equation, we start by rewriting it in standard form. The equation given is:
1x² + 1x + 2 = 3
First, we subtract 3 from both sides to set the equation to zero:
1x² + 1x + 2 – 3 = 0
This simplifies to:
x² + x – 1 = 0
Now that we have the equation in standard form, we can use the quadratic formula to find the roots. The quadratic formula is:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, a = 1, b = 1, and c = -1. We first calculate the discriminant (b² – 4ac):
Discriminant = (1)² – 4(1)(-1) = 1 + 4 = 5
Since the discriminant is positive, this means we will have two distinct real roots. Now we can substitute the values into the quadratic formula:
x = (-(1) ± √5) / (2(1))
This simplifies to:
x = (-1 ± √5) / 2
Thus, the two roots of the quadratic equation are:
x₁ = (-1 + √5) / 2
x₂ = (-1 – √5) / 2
These values are the solutions to the quadratic equation x² + x – 1 = 0.