x^2 + 15x + 36
To solve the expression x^2 + 15x + 36, we can apply the quadratic formula or factorization method, depending on the equation
To solve the expression x^2 + 15x + 36, we can apply the quadratic formula or factorization method, depending on the equation.
Method 1: Factorization
Step 1: Look for two numbers that multiply to give 36 (the constant term) and add up to 15 (the coefficient of x).
In this case, factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. We observe that 9 and 4 fit the criteria (9 + 4 = 13 and 9 * 4 = 36).
Step 2: Rewrite the equation using the two numbers found in the first step.
We can then rewrite the equation as: x^2 + 9x + 4x + 36.
Step 3: Grouping
Group the terms and factor by grouping:
(x^2 + 9x) + (4x + 36).
Factor x out of first group and 4 out of the second group:
x(x + 9) + 4(x + 9).
Step 4: Factor out the common binomial factor, (x + 9):
(x + 9)(x + 4).
Thus, the expression x^2 + 15x + 36 can be factored to (x + 9)(x + 4).
Method 2: Quadratic Formula
The quadratic formula gives us the solution for any quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are coefficients.
For the equation x^2 + 15x + 36, a = 1, b = 15, and c = 36.
The quadratic formula is given by:
x = (-b ± √(b^2 – 4ac)) / 2a.
Substitute the given values of a, b, and c into the quadratic formula:
x = (-(15) ± √((15)^2 – 4(1)(36))) / 2(1).
Simplifying further:
x = (-15 ± √(225 – 144)) / 2.
x = (-15 ± √81) / 2.
x = (-15 ± 9) / 2.
Simplifying the right side:
x = (-15 + 9)/2 = -6/2 = -3, or
x = (-15 – 9)/2 = -24/2 = -12.
Therefore, the solutions to the equation x^2 + 15x + 36 are x = -3 and x = -12.
Both methods yield the same result, and you can choose whichever method you feel more comfortable with to solve the quadratic equation.
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