Pregunta
Factor. \( 9-\left(a^{2}+2 a d+d^{2}\right) \) \( 9-\left(a^{2}+2 a d+d^{2}\right)=\square \) \( ( \) Factor completely.)
Ask by Higgins Cook. in the United States
Feb 24,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( 9 - (a^2 + 2ad + d^2) = -(a + d + 3)(a + d - 3) \)
Solución
Factor the expression \( 9-(a^2+2ad+d^2) \).
Factor the expression by following steps:
- step0: Factor:
\(9-\left(a^{2}+2ad+d^{2}\right)\)
- step1: Remove the parentheses:
\(9-a^{2}-2ad-d^{2}\)
- step2: Calculate:
\(-a^{2}-ad+3a-da-d^{2}+3d-3a-3d+9\)
- step3: Rewrite the expression:
\(-a\times a-ad+a\times 3-da-d\times d+d\times 3-3a-3d+3\times 3\)
- step4: Factor the expression:
\(-a\left(a+d-3\right)-d\left(a+d-3\right)-3\left(a+d-3\right)\)
- step5: Factor the expression:
\(\left(-a-d-3\right)\left(a+d-3\right)\)
- step6: Factor the expression:
\(-\left(a+d+3\right)\left(a+d-3\right)\)
The factored form of the expression \(9 - (a^2 + 2ad + d^2)\) is \(-(a+d+3)(a+d-3)\).
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Simplifique esta solución Beyond the Answer
To factor the expression \( 9 - \left(a^2 + 2ad + d^2\right) \), we can recognize that \( a^2 + 2ad + d^2 \) is a perfect square trinomial, which can be rewritten as \( (a + d)^2 \). Thus, your expression becomes: \[ 9 - (a + d)^2. \] Next, we can observe that this follows the difference of squares formula, which is \( A^2 - B^2 = (A - B)(A + B) \). In this case, let \( A = 3 \) (since \( 3^2 = 9 \)) and \( B = a + d \). Applying the difference of squares gives: \[ 9 - (a + d)^2 = (3 - (a + d))(3 + (a + d)). \] So the factored form is: \[ (3 - (a + d))(3 + (a + d)). \] This simplifies to: \[ (3 - a - d)(3 + a + d). \]
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