Math  /  Algebra

QuestionFind all excluded values for the expression. That is, find all values of ww for which the expression is undefined. w2+7w+10w2+4w+4\frac{w^{2}+7 w+10}{w^{2}+4 w+4}
If there is more than one value, separate them with commas.

Studdy Solution

STEP 1

1. The expression is a rational function, which is undefined when the denominator is zero.
2. We need to find the values of w w that make the denominator zero.

STEP 2

1. Identify the denominator of the expression.
2. Set the denominator equal to zero.
3. Solve the resulting equation for w w .
4. List the values of w w that make the expression undefined.

STEP 3

Identify the denominator of the expression:
The denominator is w2+4w+4 w^2 + 4w + 4 .

STEP 4

Set the denominator equal to zero:
w2+4w+4=0 w^2 + 4w + 4 = 0

STEP 5

Solve the quadratic equation w2+4w+4=0 w^2 + 4w + 4 = 0 :
This is a quadratic equation in standard form, which can be solved by factoring, completing the square, or using the quadratic formula.
Notice that w2+4w+4 w^2 + 4w + 4 can be factored as:
(w+2)(w+2)=0 (w + 2)(w + 2) = 0
or equivalently:
(w+2)2=0 (w + 2)^2 = 0

STEP 6

Solve for w w from the factored equation:
(w+2)2=0 (w + 2)^2 = 0
This implies:
w+2=0 w + 2 = 0
Solving for w w , we get:
w=2 w = -2

STEP 7

List the values of w w that make the expression undefined:
The expression is undefined for w=2 w = -2 .
The excluded value for the expression is 2 \boxed{-2} .

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