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To determine which expressions are perfect square trinomials from the given list, we need to analyze each expression and check if it can be written in the form [tex]\((ax + b)^2\)[/tex], which expands to [tex]\(a^2x^2 + 2abx + b^2\)[/tex].
1. Expression: [tex]\(x^2 + 16x + 8\)[/tex]
For this expression to be a perfect square trinomial, it should satisfy the form [tex]\( (x + b)^2 \)[/tex]. However, upon closer inspection, we can see that:
[tex]\[ (x + b)^2 = x^2 + 2bx + b^2 \][/tex]
Comparing it with [tex]\(x^2 + 16x + 8\)[/tex], we determine [tex]\(2b = 16 \implies b = 8\)[/tex] and [tex]\( b^2 \neq 8 \)[/tex], thus there is no value of [tex]\(b\)[/tex] that satisfies both conditions. Therefore, this is not a perfect square trinomial.
2. Expression: [tex]\(x^2 + 14x + 49\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = 14 \implies b = 7 \][/tex]
and
[tex]\[ b^2 = 49 \implies 7^2 = 49 \][/tex]
Both conditions hold true. Thus, this is a perfect square trinomial written as [tex]\( (x + 7)^2 \)[/tex].
3. Expression: [tex]\(x^2 - 5x + 25\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = -5 \implies b = -\frac{5}{2} \][/tex]
and
[tex]\[ b^2 = 25 \implies \left(-\frac{5}{2}\right)^2 = 6.25 \neq 25 \][/tex]
The second condition fails. Thus, this is not a perfect square trinomial.
4. Expression: [tex]\(x^2 - 24x + 144\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = -24 \implies b = -12 \][/tex]
and
[tex]\[ b^2 = 144 \implies (-12)^2 = 144 \][/tex]
Both conditions hold true. Thus, this is a perfect square trinomial written as [tex]\( (x - 12)^2 \)[/tex].
5. Expression: [tex]\(x^2 + 9x - 81\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = 9 \implies b = \frac{9}{2} \][/tex]
and
[tex]\[ b^2 = -81 \implies \left(\frac{9}{2}\right)^2 = 20.25 \neq -81 \][/tex]
The second condition fails. Thus, this is not a perfect square trinomial.
So the expressions that are perfect square trinomials are:
- [tex]\(x^2 + 14x + 49\)[/tex]
- [tex]\(x^2 - 24x + 144\)[/tex]
Hence, [tex]\(x^2 + 14x + 49\)[/tex] and [tex]\(x^2 - 24x + 144\)[/tex] are the perfect square trinomials in the given list.
1. Expression: [tex]\(x^2 + 16x + 8\)[/tex]
For this expression to be a perfect square trinomial, it should satisfy the form [tex]\( (x + b)^2 \)[/tex]. However, upon closer inspection, we can see that:
[tex]\[ (x + b)^2 = x^2 + 2bx + b^2 \][/tex]
Comparing it with [tex]\(x^2 + 16x + 8\)[/tex], we determine [tex]\(2b = 16 \implies b = 8\)[/tex] and [tex]\( b^2 \neq 8 \)[/tex], thus there is no value of [tex]\(b\)[/tex] that satisfies both conditions. Therefore, this is not a perfect square trinomial.
2. Expression: [tex]\(x^2 + 14x + 49\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = 14 \implies b = 7 \][/tex]
and
[tex]\[ b^2 = 49 \implies 7^2 = 49 \][/tex]
Both conditions hold true. Thus, this is a perfect square trinomial written as [tex]\( (x + 7)^2 \)[/tex].
3. Expression: [tex]\(x^2 - 5x + 25\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = -5 \implies b = -\frac{5}{2} \][/tex]
and
[tex]\[ b^2 = 25 \implies \left(-\frac{5}{2}\right)^2 = 6.25 \neq 25 \][/tex]
The second condition fails. Thus, this is not a perfect square trinomial.
4. Expression: [tex]\(x^2 - 24x + 144\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = -24 \implies b = -12 \][/tex]
and
[tex]\[ b^2 = 144 \implies (-12)^2 = 144 \][/tex]
Both conditions hold true. Thus, this is a perfect square trinomial written as [tex]\( (x - 12)^2 \)[/tex].
5. Expression: [tex]\(x^2 + 9x - 81\)[/tex]
Check if it matches [tex]\( (x + b)^2 = x^2 + 2bx + b^2 \)[/tex]. Here,
[tex]\[ 2b = 9 \implies b = \frac{9}{2} \][/tex]
and
[tex]\[ b^2 = -81 \implies \left(\frac{9}{2}\right)^2 = 20.25 \neq -81 \][/tex]
The second condition fails. Thus, this is not a perfect square trinomial.
So the expressions that are perfect square trinomials are:
- [tex]\(x^2 + 14x + 49\)[/tex]
- [tex]\(x^2 - 24x + 144\)[/tex]
Hence, [tex]\(x^2 + 14x + 49\)[/tex] and [tex]\(x^2 - 24x + 144\)[/tex] are the perfect square trinomials in the given list.
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