Solving the Quadratic Equation x^2 + 13x + 36: Factoring, Completing the Square, and the Quadratic Formula

x^2 + 13x + 36

To solve the quadratic equation x^2 + 13x + 36, we can use factoring, completing the square, or the quadratic formula

To solve the quadratic equation x^2 + 13x + 36, we can use factoring, completing the square, or the quadratic formula.

1. Factoring:
Since the leading coefficient is 1, we can look for two numbers whose sum is 13 and whose product is 36. The numbers that satisfy this condition are 4 and 9.
Therefore, the equation can be factored as (x + 4)(x + 9) = 0.

Setting each factor equal to zero gives us:
x + 4 = 0 or x + 9 = 0

Solving for x gives us the solutions:
x = -4 or x = -9

2. Completing the square:
To complete the square, we want to make the coefficient of the x term equal to half of the coefficient of the x term, squared. In other words, we want to rewrite the equation in the form (x + p)^2 + q = 0.

For x^2 + 13x + 36, the coefficient of the x term is 13, and half of it is 6.5. However, we can write it as 13/2 to make the calculations easier.

(x^2 + 13x) + 36 = 0
(x^2 + 13x + (13/2)^2) + 36 = (13/2)^2
(x + 13/2)^2 + 36 = 169/4

Simplifying, we get:
(x + 13/2)^2 = 169/4 – 36
(x + 13/2)^2 = 169/4 – 144/4
(x + 13/2)^2 = 25/4

Taking the square root of both sides, we get:
x + 13/2 = ±√(25/4)
x + 13/2 = ±5/2

Solving for x gives us the solutions:
x = -13/2 + 5/2 = -8/2 = -4
x = -13/2 – 5/2 = -18/2 = -9

3. Quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 – 4ac)) / 2a

For the equation x^2 + 13x + 36, we have:
a = 1, b = 13, c = 36

Plugging these values into the quadratic formula, we get:
x = (-13 ± √(13^2 – 4(1)(36))) / (2 * 1)

Simplifying the equation inside the square root:
x = (-13 ± √(169 – 144)) / 2
x = (-13 ± √(25)) / 2
x = (-13 ± 5) / 2

Solving for x gives us the solutions:
x = (-13 + 5) / 2 = -8 / 2 = -4
x = (-13 – 5) / 2 = -18 / 2 = -9

Therefore, the solutions to the equation x^2 + 13x + 36 are x = -4 and x = -9.

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