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\( x^{2} + 8x + 16 = (x + 4)^2 \)
Solución
To factor the quadratic expression \( x^{2} + 8x + 16 \), we can look for two binomials that multiply to give the original expression.
1. **Identify the coefficients:**
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 8 \) (coefficient of \( x \))
- \( c = 16 \) (constant term)
2. **Find two numbers that multiply to \( ac = 1 \times 16 = 16 \) and add up to \( b = 8 \):**
- The numbers 4 and 4 satisfy this condition because:
- \( 4 \times 4 = 16 \)
- \( 4 + 4 = 8 \)
3. **Write the factored form:**
- Since both numbers are the same, the quadratic is a perfect square.
- Therefore, \( x^{2} + 8x + 16 = (x + 4)^2 \)
**Final Answer:**
\( (x + 4)^2 \)
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Bonus Knowledge
The expression \( x^{2}+8 x+16 \) is a perfect square trinomial. It factors down to \( (x+4)^{2} \). This means that the expression represents a parabola that opens upwards with its vertex at the point (-4, 0). It's a nice friendly graph, making it easy to identify key features! If you're working with this quadratic, a common mistake is forgetting to check for other factors or solutions if you're setting it equal to zero. Remember, you can use the square root property: \( (x+4)^{2} = 0 \) leads you directly to the solution \( x = -4 \). Keep it straightforward, and don’t overlook those handy shortcuts!
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