Question
a) \( y^{2}-y-30 \)
Ask by Hill Gray. in El Salvador
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y^{2} - y - 30 = (y + 5)(y - 6) \)
Solution
1. Planteamos el trinomio:
\[
y^2 - y - 30
\]
2. Buscamos dos números \(a\) y \(b\) tales que:
\[
a + b = -1 \quad \text{y} \quad a \cdot b = -30.
\]
Probamos con \(a = 5\) y \(b = -6\), ya que:
\[
5 + (-6) = -1 \quad \text{y} \quad 5 \cdot (-6) = -30.
\]
3. Descomponemos el trinomio usando estos números:
\[
y^2 - y - 30 = y^2 + 5y - 6y - 30.
\]
4. Factorizamos por agrupación:
\[
\begin{aligned}
y^2 + 5y - 6y - 30 &= y(y + 5) - 6(y + 5) \\
&= (y + 5)(y - 6).
\end{aligned}
\]
5. Por lo tanto, el trinomio se factoriza como:
\[
y^2 - y - 30 = (y + 5)(y - 6).
\]
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Simplify this solution Bonus Knowledge
To factor the quadratic expression \( y^2 - y - 30 \), we look for two numbers that multiply to \(-30\) and add to \(-1\). Those numbers are \(5\) and \(-6\). Thus, we can write the expression as \((y - 6)(y + 5)\). This means the solutions to \( y^2 - y - 30 = 0 \) are \( y = 6 \) and \( y = -5 \). If you want to solve quadratics effortlessly, always remember the “AC method” where you multiply \(a\) and \(c\) in \(ax^2 + bx + c\) to find two numbers whose product is \(ac\) and sum is \(b\). It's a neat trick that can save you time on tests and homework!
