Factor completely: $25f^2 - 30f + 9$
Understand the Problem
The question asks to factorize a quadratic expression, this involves expressing the quadratic as a product of its factors.
Answer
$(5f - 3)^2$
Answer for screen readers
$(5f - 3)^2$
Steps to Solve
- Recognize the perfect square trinomial pattern
The given expression $25f^2 - 30f + 9$ looks like a perfect square trinomial of the form $a^2 - 2ab + b^2$, which factors to $(a-b)^2$.
- Identify 'a' and 'b'
We want to find $a$ and $b$ such that: $a^2 = 25f^2$ and $b^2 = 9$. Taking the square root of both sides, we get: $a = 5f$ and $b = 3$.
- Verify the middle term
Check if the middle term $-30f$ matches $-2ab$: $-2ab = -2(5f)(3) = -30f$. Since the middle term matches, the given trinomial is indeed a perfect square trinomial.
- Factor the trinomial
Using the form $(a-b)^2$, we substitute $a = 5f$ and $b = 3$: $(5f - 3)^2$
$(5f - 3)^2$
More Information
The expression $25f^2 - 30f + 9$ is a perfect square trinomial, which is a special type of quadratic expression that can be factored into the square of a binomial.
Tips
A common mistake is to incorrectly identify 'a' and 'b' when recognizing the perfect square trinomial pattern. Another mistake could be factoring it as $(5f + 3)^2$ instead of $(5f - 3)^2$. Remember to pay attention to the sign of the middle term to determine whether the binomial is a sum or a difference.
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