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

Comments (7) on “Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. c^2+2c”

1. [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]

   Reply

2. 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].

   Reply

3.  [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]

   Reply

4. (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

   Reply

5. (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

   Reply

6. 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]

   Reply

7. 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

   Reply