Question - Solving a Quadratic Equation by Factoring
Solution:
The expression in the image is a quadratic equation:\[ 3n^2 + 9n - 30 = 0 \]To solve this quadratic equation, we can first attempt to factor it if it is factorable. If it's not easily factorable, we would use the quadratic formula.Let's see if the equation is factorable by looking for two numbers that multiply to $$3 \times (-30) = -90$$ and add up to $$9$$. These two numbers are 15 and -6 because $$15 \times -6 = -90$$ and $$15 + (-6) = 9$$.We can then express $$9n$$ as $$15n - 6n$$, rewriting the equation as:\[ 3n^2 + 15n - 6n - 30 = 0 \]Then we group the terms:\[ (3n^2 + 15n) - (6n + 30) = 0 \]Factoring by grouping, we take out the greatest common factor from each group:\[ 3n(n + 5) - 6(n + 5) = 0 \]Now, since both terms have a common factor of $$(n + 5)$$, we factor it out:\[ (n + 5)(3n - 6) = 0 \]Setting each factor equal to zero gives us our potential solutions:\[ n + 5 = 0 \quad \Rightarrow \quad n = -5 \]\[ 3n - 6 = 0 \quad \Rightarrow \quad 3n = 6 \quad \Rightarrow \quad n = 2 \]Therefore, the solutions to the quadratic equation are $$n = -5$$ and $$n = 2$$.
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