Solving the Quadratic Equation: x² + 2x – 7 = 0
To solve the quadratic equation x² + 2x – 7 = 0, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, we can identify the coefficients as follows:
- a = 1 (the coefficient of x²)
- b = 2 (the coefficient of x)
- c = -7 (the constant term)
Now, we need to calculate the discriminant (b² – 4ac):
b² = 2² = 4
4ac = 4 * 1 * (-7) = -28
So,
b² – 4ac = 4 + 28 = 32
Since the discriminant is positive, we will have two distinct real solutions. Now we plug the values into the quadratic formula:
x = (-2 ± √32) / (2 * 1)
Simplifying further:
x = (-2 ± 4√2) / 2
By simplifying this, we get:
x = -1 ± 2√2
Thus, the two solutions are:
- x₁ = -1 + 2√2 (approximately 1.828)
- x₂ = -1 – 2√2 (approximately -3.828)
In conclusion, the solutions to the equation x² + 2x – 7 = 0 are:
- x ≈ 1.828
- x ≈ -3.828