Factoring and Completing the Square Methods for x^2 + 10x + 16: Math Explained

x^2 + 10x + 16

To work with the expression x^2 + 10x + 16, we can either factor it or complete the square

To work with the expression x^2 + 10x + 16, we can either factor it or complete the square.

1. Factoring:
To factor the expression, we need to find two numbers whose sum is 10 and whose product is 16. Notice that the factors of 16 are 1 and 16, as well as 2 and 8. However, neither of these pairs sum up to 10.

Since the expression cannot be factored using whole numbers, we can try factoring it using irrational numbers. We can use the quadratic formula to find the roots of the quadratic equation and then factor it accordingly.

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

In our case, the quadratic equation is x^2 + 10x + 16, which can be rewritten as:
x = (-10 ± √(10^2 – 4(1)(16))) / (2(1))

Simplifying the equation further:
x = (-10 ± √(100 – 64)) / 2
x = (-10 ± √36) / 2
x = (-10 ± 6) / 2

Now, we have the two possible values for x:
x = (-10 + 6) / 2 = -4 / 2 = -2
x = (-10 – 6) / 2 = -16 / 2 = -8

Therefore, the expression x^2 + 10x + 16 can be factored as (x + 2)(x + 8).

2. Completing the square:
To complete the square for the quadratic expression x^2 + 10x + 16, we need to follow these steps:

Step 1: Divide the coefficient of x by 2 and square it:
(10 / 2)^2 = 25

Step 2: Add the result from Step 1 to both sides of the expression:
x^2 + 10x + 25 + 16 = (x^2 + 10x + 25) + 16

Step 3: Rewrite the left side as a perfect square trinomial and simplify:
(x + 5)^2 + 16 = x^2 + 10x + 25 + 16 = x^2 + 10x + 41

Now, the expression x^2 + 10x + 16 can be written as (x + 5)^2 + 41.

Both factoring and completing the square methods provide equivalent forms for the quadratic expression.

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