Question

Write the coefficient of $y$ in the expansion of ${\left( {5 - y} \right)^2}$.

Answer 1

Hint: We have given expansion${\left( {5 - y} \right)^2}$. We have to find the coefficient of $y$ in the expansion. For this, we have to expand the expression. The expression is in the form ${\left( {a - b} \right)^2}$. So, we expand as ${\left( {a - b} \right)^2} = {a^2} + {b^2} + 2ab$ with the help of this we can find the coefficient of $y$.

Complete step-by-step answer:
The given expansion is ${\left( {5 - y} \right)^2}$
The expansion is in the form ${\left( {a - b} \right)^2}$
We apply the formula ${\left( {a - b} \right)^2} = {a^2} + {b^2} + 2ab$
$a = 5,b = y$
So, ${\left( {5 - y} \right)^2} = {\left( 5 \right)^2} + {\left( y \right)^2} - 2*5*y$
$\Rightarrow$ ${\left( {5 - y} \right)^2} = 25 + {y^2} - 10y$
So, we have ${\left( {5 - y} \right)^2} = 25 + {y^2} - 10y$
In $25 + {y^2} - 10y$, coefficient of $y$ is $ - 10$
So, the coefficient of ${\left( {5 - y} \right)^2}$ is $ - 10$.

Note: In Algebra, a quadratic equation is any equation that can be rearranged in standard form $a{x^2} + bx + c = 0$ where x is an unknown variable and $a,b$ and $c$ are known variables.
$a \ne 0$. if $a = 0$ then the equation is not a quadratic equation. $a,b$ and $c$ are coefficients of an equation and may be distinguished by calling them quadratic coefficient.
The value of ‘$x$’ that satisfies the equation is called the solution of the equation. The quadratic equation has at most two solutions. If there is no real solution, the solution is called a complex solution, if there is only one root then it is called double solution.
There are three basic methods of solving quadratic equations: factorising, using quadratic formulas and completing square methods.
Factorising method:
To solve the equation by factorising method
(i)Put all the terms on one side of the equal sign, leaving zero on the other side.
Factor
(ii)Set each factor equal to zero.
(iii)Solve each of these equations.