Find the value of k that makes a trinomial a perfect square

Expert intro: To make a trinomial a perfect square, the middle term has to match the pattern from a binomial square. That gives a direct equation for $k$ once you compare coefficients.

Detailed walkthrough

A perfect square trinomial has the form

$(ax+b)^2=a^2x^2+2abx+b^2$

a) $25y^2+ky+144$

Here,

• $25y^2=(5y)^2$
• $144=12^2$

So the middle term must be

$2(5y)(12)=120y$

Hence,

$\boxed{k=120}$

b) $9x^2-42x+k$

Here,

• $9x^2=(3x)^2$
• middle term should be $2(3x)(b)=-42x$

So

$6b=-42 \Rightarrow b=-7$

Then the constant term is

$k=b^2=(-7)^2=49$

So,

$\boxed{k=49}$

💡 Pitfall guide

Don’t solve these by guessing the factorization first. It’s easier and safer to match the middle coefficient with $2ab$. Also remember that the constant term is always the square of the second binomial term, so it must be nonnegative.

🔄 Real-world variant

If the middle term had been positive in part (b), the same method would still work, but the sign of $b$ would change. The constant term would stay the same because squaring removes the sign.

🔍 Related terms

perfect square trinomial, coefficient matching, binomial square