Solve the quadratic equation. Show all of your steps.
x^2 + 3x - 5 = 0
To solve the quadratic equation x^2 + 3x - 5 = 0, we can use either factoring, completing the square, or the quadratic formula. Let's use the quadratic formula.
The quadratic formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions (or roots) can be found using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In our equation, a = 1, b = 3, and c = -5. Plugging these values into the quadratic formula:
x = (-(3) ± √((3)^2 - 4(1)(-5)))/(2(1))
Simplifying further:
x = (-3 ± √(9 + 20))/2
x = (-3 ± √29)/2
Therefore, the quadratic equation x^2 + 3x - 5 = 0 has two solutions:
x = (-3 + √29)/2
x = (-3 - √29)/2