# How do you determine whether 4x^2-3x+9 is a perfect square trinomial?

Jul 2, 2017

$4 {x}^{2} - 3 x + 9$ is not a perfect trinomial.

#### Explanation:

Let us discuss the identity ${\left(a \pm b\right)}^{2} = {a}^{2} \pm 2 a b + {b}^{2}$

Observe the RHS. While

• ${a}^{2}$ is the square of the first term $a$
• ${b}^{2}$ is the square of the second term $b$
• sign of ${a}^{2}$ and ${b}^{2}$ is same and is positive
• $2 a b$ is twice the product of two terms $a$ and $b$ and its sign is $+$ or $-$ depending on its sign in ${\left(a \pm b\right)}^{2}$

In $4 {x}^{2} - 3 x + 9$, while $4 {x}^{2} = {\left(2 x\right)}^{2}$ and $9 = {3}^{2}$,

but (leaving the sign) middle term is not $2 \times 2 x \times 3$ which is $12 x$ but is $3 x$

Hence, $4 {x}^{2} - 3 x + 9$ is not a perfect trinomial.