Question

How do you find the product of \[{\left( {x + 7} \right)^2}\]?

Answer 1

Hint: The above question is based on the concept of multiplication of polynomials by binomials. The main approach towards solving the equation is applying the formula of the square of the sum of terms which will result in the terms in power of 2 and then further calculating the quadratic equation we get the value of x.

Complete step by step answer:
The above expression is based on multiplication of polynomials. Polynomial is an expression which consists of coefficients and variables. And binomial is a polynomial with two terms.The above given expression is \[{\left( {x + 7} \right)^2}\]. This expression can be split into two terms in the following way.
\[\left( {x + 7} \right)\left( {x + 7} \right)\]
We split it into two terms because the highest power mentioned is two that is why we split the two polynomials. Now by applying the formula of square of sum of terms,
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
So further we get is,
\[\Rightarrow {\left( {x + 7} \right)^2} = {x^2} + 2x \times 7 + {7^2} \\
\Rightarrow {x^2} + 14x + 49 = 0 \\ \]
Since we get the quadratic equation, we will further solve the factors.First step is by multiplying the coefficient of \[{x^2}\] and the constant term 49, we get \[49{x^2}\].After this, factors of \[49{x^2}\]should be calculated in such a way that their addition should be equal to 14x.Factors of 49 can be 7 and 7 and equating with zero.
\[\Rightarrow {x^2} + 7x + 7x + 49 = 0 \\
\Rightarrow x(x + 7) + 7(x + 7) = 0 \\
\Rightarrow (x + 7)(x + 7) = 0 \\
\therefore x = - 7,x = - 7 \]

Note: An important thing to note is that in quadratic equation, an alternative way of finding the factors is by using a formula which is given \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].By substituting the values of a=1 ,b=14,c=49 we get \[x = \dfrac{{ - 14 \pm \sqrt {{{\left( {14} \right)}^2} - 4\left( 1 \right)\left( {49} \right)} }}{{2\left( 1 \right)}}\] and we get the value as \[x = - 7, - 7\].