Analyzing a Quadratic Expression: Factoring and Finding Roots of x^2 + 13x + 36

x^2 + 13x + 36

To understand the expression x^2 + 13x + 36, we need to recognize that it is a quadratic expression, also known as a quadratic polynomial

To understand the expression x^2 + 13x + 36, we need to recognize that it is a quadratic expression, also known as a quadratic polynomial. A quadratic expression consists of three terms, with the highest power being 2.

Here, the expression is in the form of Ax^2 + Bx + C, where A, B, and C are coefficients.

In this case:
A = 1
B = 13
C = 36

To further analyze the expression, we may want to determine if it can be factored, if its roots can be found using the quadratic formula, or if it represents a perfect square trinomial.

Let’s check if it can be factored.
To factor a quadratic expression, we need to find two binomials whose product equals the quadratic expression. In this case, we need to find two binomials of the form (x + ?)(x + ?) that multiply to give x^2 + 13x + 36.
We are looking for two numbers whose product is 36 and whose sum is 13.

Let’s find the factors of 36 and check if any of them can be added up to 13:
1 x 36 = 36
2 x 18 = 36
3 x 12 = 36
4 x 9 = 36
6 x 6 = 36

The pair that adds up to 13 is 4 and 9. Therefore, we can factorize the quadratic expression as (x + 4)(x + 9).

So, x^2 + 13x + 36 = (x + 4)(x + 9).

Factoring a quadratic expression allows us to find the roots of the equation, which are the values of x that make the expression equal to zero. In this case, we can set the factored expression equal to zero and solve for x:

(x + 4)(x + 9) = 0

Setting each factor equal to zero and solving:

x + 4 = 0 –> x = -4
x + 9 = 0 –> x = -9

Therefore, this quadratic expression has two roots: x = -4 and x = -9.

By analyzing the expression x^2 + 13x + 36, we were able to factor it as (x + 4)(x + 9) and find its roots as x = -4 and x = -9.

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