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\( y^{2} - 8y + 15 = (y - 3)(y - 5) \)
Solución
To factor the quadratic expression \( y^{2} - 8y + 15 \), follow these steps:
1. **Identify the coefficients:**
- The quadratic is in the form \( ay^{2} + by + c \), where:
- \( a = 1 \)
- \( b = -8 \)
- \( c = 15 \)
2. **Find two numbers that multiply to \( ac = 1 \times 15 = 15 \) and add up to \( b = -8 \):**
- The numbers are **-3** and **-5** because:
- \( (-3) \times (-5) = 15 \)
- \( (-3) + (-5) = -8 \)
3. **Rewrite the middle term using these numbers:**
\[
y^{2} - 3y - 5y + 15
\]
4. **Factor by grouping:**
\[
(y^{2} - 3y) - (5y - 15)
\]
\[
y(y - 3) - 5(y - 3)
\]
5. **Factor out the common binomial factor \((y - 3)\):**
\[
(y - 3)(y - 5)
\]
**Final Factored Form:**
\[
y^{2} - 8y + 15 = (y - 3)(y - 5)
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Extra Insights
To factor the quadratic expression \( y^{2} - 8y + 15 \), we look for two numbers that multiply to 15 (the constant term) and add up to -8 (the coefficient of the linear term). Those two numbers are -3 and -5. Therefore, we can rewrite the expression as \( (y - 3)(y - 5) \). Thus, the factored form is: \[ (y - 3)(y - 5) \] Now, let’s find the roots of this equation, which are simply the values of \( y \) that make either factor equal to zero. By setting each factor to zero, we find \( y = 3 \) and \( y = 5 \). This tells us that the parabola defined by the quadratic equation touches the x-axis at these points, providing insight into its graph!
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