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Pls help this is really hard

Directions: Complete each trinomial so that it is a perfect square​

Pls help this is really hard Directions: Complete each trinomial so that it is a perfect-example-1
User FaNaJ
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1 Answer

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Answer:


\large\text{1.\quad$x^2+18x+\boxed{81}$}


\large\text{2.\quad$x^2+\boxed{4}\:x+4$}


\large\text{3.\quad$x^2-100x+\boxed{2500}$}

Explanation:

A perfect square trinomial is a type of quadratic trinomial (a polynomial with three terms) that can be factored into the square of a binomial expression in one of two forms: (ax + b)² or (ax - b)².

To complete each of the given trinomials so that they become perfect square trinomials, they should be in the form:

(ax + b)² = a²x² + 2abx + b²

(ax + b)² = a²x² - 2abx + b²


\hrulefill

Question 1

Given:


x^2 + 18x + \boxed{\phantom{w}}

Since both operations are addition, we use the form (ax + b)².

Compare the coefficients of the given expression with a²x² + 2abx + b²:


a^2 = 1 \implies a=1


2ab = 18

Substituting a = 1 into 2ab = 18 and solving for b:


\begin{aligned}2 \cdot 1 \cdot b&=18\\2b&=18\\b&=9\end{aligned}

As b = 9, then:


b^2=9^2=81

Therefore, the completed perfect square trinomial is:


\large\text{$x^2 + 18x + \boxed{81}$}

This trinomial can be factored as (x + 9)².


\hrulefill

Question 2

Given:


x^2 + \boxed{\phantom{w}}\:x + 4

Since both operations are addition, we use the form (ax + b)².

Compare the coefficients of the given expression with a²x² + 2abx + b²:


a^2 = 1 \implies a=1


b^2=4\implies b=2

Substituting a = 1 and b = 2 into 2ab:


2ab=2(1)(2)=4

Therefore, the completed perfect square trinomial is:


\large\text{$x^2 + \boxed{4}\:x + 4$}

This trinomial can be factored as (x + 2)².


\hrulefill

Question 3

Given:


x^2 - 100x + \boxed{\phantom{w}}

Since the first operation is subtraction and the second operation is addition, we use the form (ax - b)².

Compare the coefficients of the given expression with a²x² - 2abx + b²:


a^2 = 1 \implies a=1


2ab = 100

Substituting a = 1 into 2ab = 100 and solving for b:


\begin{aligned}2 \cdot 1 \cdot b&=100\\2b&=100\\b&=50\end{aligned}

As b = 50, then:


b^2=50^2=2500

Therefore, the completed perfect square trinomial is:


\large\text{$x^2 -100x + \boxed{2500}$}

This trinomial can be factored as (x - 50)².

User Derrops
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