Factor completely: 144n^2 - 168n + 49
Understand the Problem
The question asks to factor the given quadratic expression completely. This involves rewriting the expression as a product of two binomials. We'll need to identify the coefficients and constants, and then find factors that satisfy the necessary conditions for factoring the quadratic.
Answer
$(12n - 7)^2$
Answer for screen readers
$(12n - 7)^2$
Steps to Solve
- Recognize the pattern
Notice that $144n^2 = (12n)^2$ and $49 = (7)^2$. This suggests that the given expression might be a perfect square trinomial of the form $(a - b)^2 = a^2 - 2ab + b^2$.
- Check if it's a perfect square trinomial
If $a = 12n$ and $b = 7$, then $2ab = 2(12n)(7) = 168n$. The middle term of the given expression is $-168n$, which matches $-2ab$. So the given expression is indeed a perfect square trinomial.
- Factor the perfect square trinomial
Since $144n^2 - 168n + 49$ fits the pattern $a^2 - 2ab + b^2$ with $a = 12n$ and $b = 7$, we can factor it as $(a - b)^2$.
- Write the factored form
Substituting $a = 12n$ and $b = 7$, we get $(12n - 7)^2$.
$(12n - 7)^2$
More Information
The given quadratic expression is a perfect square trinomial, which makes it easier to factor. Recognizing these patterns simplifies the factoring process.
Tips
A common mistake would be to incorrectly identify $a$ and $b$, or to miscalculate $2ab$, leading to an incorrect factorization. Another mistake could be not recognizing the perfect square trinomial pattern at all and attempting a more complicated factoring method.
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