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3. Which equation contains a perfect square trinomial?

A. [tex]x^2 + 14x + 49 = 0[/tex]
B. [tex]x^2 - 5x + 64 = 0[/tex]
C. [tex]x^2 - 6x + 72 = 0[/tex]
D. [tex]x^2 + 2x - 4 = 0[/tex]

Sagot :

GinnyAnswer
To determine which of the given equations contains a perfect square trinomial, we need to understand what a perfect square trinomial is. A quadratic expression [tex]\(ax^2 + bx + c\)[/tex] is a perfect square trinomial if it can be written in the form [tex]\((ax + b)^2\)[/tex]. A common characteristic of a perfect square trinomial is that the constant term [tex]\( c \)[/tex] is a perfect square, and [tex]\( b \)[/tex] is twice the product of [tex]\( a \)[/tex] and the square root of [tex]\( c \)[/tex].

Let's analyze each equation step-by-step:

1. Equation 1: [tex]\(x^2 + 14x + 49 = 0\)[/tex]
We rewrite it as:
[tex]\[ x^2 + 14x + 49 \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 14\)[/tex], and [tex]\(c = 49\)[/tex].

We check if:
[tex]\[ b^2 = 4ac \][/tex]
[tex]\[ 14^2 = 4 \cdot 1 \cdot 49 \][/tex]
[tex]\[ 196 = 196 \][/tex]
This equation is a perfect square trinomial because it can be factored as:
[tex]\[ (x + 7)^2 = 0 \][/tex]

2. Equation 2: [tex]\(x^2 - 5x + 64 = 0\)[/tex]
We rewrite it as:
[tex]\[ x^2 - 5x + 64 \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 64\)[/tex].

We check if:
[tex]\[ b^2 = 4ac \][/tex]
[tex]\[ (-5)^2 = 4 \cdot 1 \cdot 64 \][/tex]
[tex]\[ 25 \neq 256 \][/tex]
This equation is not a perfect square trinomial.

3. Equation 3: [tex]\(x^2 - 6x + 72 = 0\)[/tex]
We rewrite it as:
[tex]\[ x^2 - 6x + 72 \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = 72\)[/tex].

We check if:
[tex]\[ b^2 = 4ac \][/tex]
[tex]\[ (-6)^2 = 4 \cdot 1 \cdot 72 \][/tex]
[tex]\[ 36 \neq 288 \][/tex]
This equation is not a perfect square trinomial.

4. Equation 4: [tex]\(x^2 + 2x - 4 = 0\)[/tex]
We rewrite it as:
[tex]\[ x^2 + 2x - 4 \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -4\)[/tex].

We check if:
[tex]\[ b^2 = 4ac \][/tex]
[tex]\[ 2^2 = 4 \cdot 1 \cdot (-4) \][/tex]
[tex]\[ 4 \neq -16 \][/tex]
This equation is not a perfect square trinomial.

Based on these calculations, we find that only the first equation:
[tex]\[ x^2 + 14x + 49 = 0 \][/tex]
contains a perfect square trinomial.

So, the answer is the first equation.
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