Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. c^2+2c

Skip to content# Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. c^2+2c

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. c^2+2c

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[tex]n^2 + \frac{5}{2}n + \frac{25}{16}[/tex]

[tex](n + \frac{5}{4})^2[/tex]

Step-by-step explanation:

Given

[tex]n^2 + \frac{5}{2}n[/tex]

Required

(a) Make a perfect square trinomial

(b) Write as binomial square

Solving (a)

Let the missing part of the expression be k;

This gives

[tex]n^2 + \frac{5}{2}n + k[/tex]

To solve for k, we need to square half the coefficient of n;

i.e. Since the coefficient of n is [tex]\frac{5}{2}[/tex], then

[tex]k = (\frac{1}{2} * \frac{5}{2})^2[/tex]

[tex]k = (\frac{5}{4})^2[/tex]

[tex]k = \frac{25}{16}[/tex]

Hence;

[tex]n^2 + \frac{5}{2}n + k[/tex] = [tex]n^2 + \frac{5}{2}n + \frac{25}{16}[/tex]

Solving (b)

[tex]n^2 + \frac{5}{2}n + \frac{25}{16}[/tex]

Expand [tex]\frac{5}{2}n[/tex]

[tex]n^2 + \frac{5}{4}n+ \frac{5}{4}n + \frac{25}{16}[/tex]

Factorize

[tex]n(n + \frac{5}{4})+ \frac{5}{4}(n + \frac{5}{4})[/tex]

[tex](n + \frac{5}{4})(n + \frac{5}{4})[/tex]

[tex](n + \frac{5}{4})^2[/tex]

Hence:

[tex]n^2 + \frac{5}{2}n + \frac{25}{16}[/tex] = [tex](n + \frac{5}{4})^2[/tex]

Perfect square trinomial: [tex]n^2+n+\dfrac{1}{4}[/tex]

Binomial squared: [tex]\left(n+\dfrac{1}{2}\right)^2[/tex].

Step-by-step explanation:

The given expression is [tex]n^2+n.[/tex]

Here coefficient of n is 1.

Now, find square of half of coefficient of n.

[tex]\left(\dfrac{1}{2}\times 1\right)^2=\dfrac{1}{4}[/tex]

Now, add [tex]\dfrac{1}{4}[/tex] in the expression to make it perfect square trinomial.

[tex]n^2+n+\dfrac{1}{4}[/tex]

[tex]\Rightarrow n^2+n+\left(\dfrac{1}{2}\right)^2[/tex]

[tex]\Rightarrow \left(n+\dfrac{1}{2}\right)^2[/tex] [tex]\because (a+b)^2=a^2+2ab+b^2[/tex]

Therefore, perfect square trinomial is [tex]n^2+n+\dfrac{1}{4}[/tex] and binomial squared is [tex]\left(n+\dfrac{1}{2}\right)^2[/tex].

[tex]\bigg(n+\dfrac{5}{4}\bigg)^2[/tex]

Step-by-step explanation:

[tex]n^2+\dfrac{5}{2}n+\underline{\qquad}\\\\\\n^2+\dfrac{5}{2}n+\bigg(\dfrac{5}{2\cdot 2}\bigg)^2\\\\\\n^2+\dfrac{5}{2}n+\bigg(\dfrac{5}{4}\bigg)^2\\\\\\=\bigg(n+\dfrac{5}{4}\bigg)^2[/tex]

(n- 2/3)²

Step-by-step explanation:

Perfect square trinomial is: a²+2ab+b²= (a+b)²

We have:

n² - 4n/3

It can be put as:

n² -2×n×2/3

Here we consider n = a and -2/3 = b, then

b²= (-2/3)²= 4/9

Now we add 4/9 to a given binomial to make it perfect square:

n² - 2×n×3/2 + 4/9= (n- 2/3)²

So, added 4/9 and got a perfect square (n- 2/3)²

(y+5)²

Step-by-step explanation:

Every perfect square trinomial, completes the rule:

(a+b)²=(a²+2ab+b²)

Let's complete the PST:

(a) would be the square root of y², which is y.

(b) would be a number that multiplied by 2y gives us 10y, which is 5.

Our perfect square trinomial is y²+10y+25

Finally, using the mentioned rule, the binomial square for that equation is (y+5)²

Take the coefficient of ''n'', divide it by 2, and raise it to the second power;

[tex]n^2+n+(\frac{1}{2})^2[/tex]

[tex]n^2+n+\frac{1}{4}=0[/tex]

[tex]=(n+\frac{1}{2} )^2[/tex]

Step-by-step explanation:

Hello, please consider the following.

[tex]q^2-\dfrac{1}{4}q=q^2-2\cdot \dfrac{1}{8}\cdot x=(q-\dfrac{1}{8})^2-\dfrac{1}{8^2}\\\\=\boxed{(q-\dfrac{1}{8})^2-\dfrac{1}{64}}\\\\\text{ We need to add } \dfrac{1}{64} \text{ to complete the square.}\\\\[/tex]

[tex]\Large \boxed{\sf \ \ q^2-\dfrac{1}{4}q+\dfrac{1}{64}=(q-\dfrac{1}{8})^2 \ \ }[/tex]

Hope this helps.

Do not hesitate if you need further explanation.

Thank you