Pregunta
11) \( x^{2}+4 x y-12 y^{2} \)
Ask by Burton Hilton. in the United States
Mar 11,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
\( x^{2} + 4xy - 12y^{2} = (x + 6y)(x - 2y) \)
Solución
1. Rewrite the expression:
\[
x^2 + 4xy - 12y^2
\]
2. We need to factor it in the form:
\[
(x + my)(x + ny)
\]
When expanded, this becomes:
\[
x^2 + (m+n)xy + mny^2
\]
3. Compare coefficients with the given expression:
\[
x^2 + 4xy - 12y^2
\]
We have:
\[
m+n = 4 \quad \text{and} \quad m \cdot n = -12
\]
4. Find two numbers \( m \) and \( n \) that satisfy these conditions. One valid pair is \( m = 6 \) and \( n = -2 \) since:
\[
6 + (-2) = 4 \quad \text{and} \quad 6 \times (-2) = -12
\]
5. Therefore, the factorization is:
\[
(x+6y)(x-2y)
\]
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución The Deep Dive
This quadratic expression can be factored with some creative algebra! It resembles the standard form of a quadratic equation, which can hint at factoring. The expression \( x^{2}+4xy-12y^{2} \) can be factored into \( (x - 2y)(x + 6y) \) by looking for two numbers that multiply to the product of the first coefficient and the constant (-12) while adding up to the middle coefficient (4). And there you have it: a neat product of two binomials, making it easier to solve for values of \( x \) when set to zero. Always remember, factoring can simplify your equations and make them more manageable!
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