Question

Solve the quadratic equation.
x^2 + 3x - 5 = 0

Answer 1

To solve the quadratic equation x^2 + 3x - 5 = 0, we can either use factoring, completing the square, or the quadratic formula.

Factoring:
To factor the quadratic equation, we need to find two numbers that multiply to give -5 (the constant term), and add up to 3 (the coefficient of the x term). The numbers are 5 and -1.

Therefore, we can write the equation as:
(x + 5)(x - 1) = 0

Setting each factor equal to zero and solving for x, we get:
x + 5 = 0 or x - 1 = 0
x = -5 or x = 1

Completing the square:
A quadratic equation in the form of x^2 + bx + c = 0 can be rewritten as (x + (b/2))^2 = -(c - (b/2)^2).

For the equation x^2 + 3x - 5 = 0, we complete the square by adding ((3/2)^2) = (9/4) to both sides:
x^2 + 3x - 5 + (9/4) = (9/4)
(x + (3/2))^2 = 29/4

Taking the square root of both sides, we get:
x + (3/2) = ± sqrt(29)/2

Solving for x, we have:
x = -3/2 ± sqrt(29)/2
This can also be written as:
x = (-3 ± sqrt(29))/2

Quadratic formula:
The quadratic formula states that for a quadratic equation in the form of ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a

For the equation x^2 + 3x - 5 = 0, a = 1, b = 3, and c = -5.
Plugging these values into the quadratic formula, we have:
x = (-(3) ± sqrt((3)^2 - 4(1)(-5))) / (2(1))
x = (-3 ± sqrt(9 + 20)) / 2
x = (-3 ± sqrt(29)) / 2

This also gives us the solutions:
x = (-3 + sqrt(29))/2 or x = (-3 - sqrt(29))/2