The fraction to be added toy2−10y/11+11/121make it a perfect square is:
A14/121
B11/121
C14/11
D15/11
Answer:
A. 14/121
Read Explanation:
Perfect Square Trinomials in Algebra
A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form of a perfect square trinomial is $ax^2 + bx + c$. To form a perfect square trinomial from a binomial, we often need to add a specific constant term.
Completing the Square Method
The method of completing the square is used to transform a quadratic expression into a perfect square trinomial.
Consider a quadratic expression in the form $x^2 + Bx$. To make it a perfect square, we need to add $\left(\frac{B}{2}\right)^2$.
The resulting trinomial will be $x^2 + Bx + \left(\frac{B}{2}\right)^2$, which factors as $\left(x + \frac{B}{2}\right)^2$.
Applying to the Given Expression
The given expression is $y^2 - \frac{10y}{11} + \frac{11}{121}$.
We can rewrite the expression as $y^2 + \left(-\frac{10}{11}\right)y + \frac{11}{121}$.
Here, the coefficient of $y$ (which corresponds to $B$ in the general form) is $-\frac{10}{11}$.
To complete the square, we need to find $\left(\frac{B}{2}\right)^2$.
Calculate $\frac{B}{2}$: $\frac{-\frac{10}{11}}{2} = -\frac{10}{2 \times 11} = -\frac{5}{11}$.
Now, square this value: $\left(-\frac{5}{11}\right)^2 = \frac{(-5)^2}{(11)^2} = \frac{25}{121}$.
The term needed to make $y^2 - \frac{10y}{11}$ a perfect square is $\frac{25}{121}$.
The expression $y^2 - \frac{10y}{11} + \frac{25}{121}$ is a perfect square trinomial, factoring to $\left(y - \frac{5}{11}\right)^2$.
Determining the Fraction to Add
The original expression is $y^2 - \frac{10y}{11} + \frac{11}{121}$.
We have identified that the term required for a perfect square is $\frac{25}{121}$.
The current constant term is $\frac{11}{121}$.
The fraction that needs to be added is the difference between the required term and the current term: $\frac{25}{121} - \frac{11}{121}$.
Calculate the difference: $\frac{25 - 11}{121} = \frac{14}{121}$.