Question
Factoviser \( (x+4)^{2}-16 \)
Ask by Logan Rogers. in Mauritania
Dec 30,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
\( (x + 4)^2 - 16 = x(x + 8) \)
Solution
Pour factoriser l'expression \( (x + 4)^2 - 16 \), suivez les étapes suivantes :
1. **Reconnaître une différence de carrés** :
L'expression peut être réécrite comme une différence de deux carrés :
\[
(x + 4)^2 - 4^2
\]
2. **Appliquer la formule de la différence de carrés** :
La formule générale pour la différence de carrés est :
\[
a^2 - b^2 = (a - b)(a + b)
\]
En appliquant cette formule à notre expression :
\[
(x + 4)^2 - 4^2 = (x + 4 - 4)(x + 4 + 4)
\]
3. **Simplifier les termes** :
\[
(x + 4 - 4) = x
\]
\[
(x + 4 + 4) = x + 8
\]
4. **Écrire la factorisation finale** :
\[
(x + 4)^2 - 16 = x(x + 8)
\]
**Résultat :**
\[
(x + 4)^2 - 16 = x(x + 8)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
Let's first simplify the expression \( (x+4)^{2}-16 \). This is a difference of squares, which can be expressed as \( (a^2 - b^2) = (a-b)(a+b) \). Here, \( a = (x+4) \) and \( b = 4 \) (since \( 16 = 4^2 \)). So we rewrite it as: \[ (x+4 - 4)(x+4 + 4) = (x)(x+8) = x(x+8). \] Now the expression is factored as \( x(x + 8) \). Factoring is a powerful tool in algebra that simplifies expressions and solves equations more easily. Using this factorization, we can find the roots of the equation by setting it equal to zero: \( x(x + 8) = 0 \), leading to solutions of \( x = 0 \) and \( x = -8 \). This highlights how factoring helps in solving quadratic equations efficiently!
