Question

How do you solve it ${{\left( x+2 \right)}^{2}}=9$ ?

Answer 1

Hint: ${{\left( x+2 \right)}^{2}}=9$ is the equation which is needed to be solved and find the value of $x$ . Find out that the equation is in which type of format. After checking the format of the equation, find the formula which is suitable to find the value of $x$.

Complete step-by-step solution:
The given equation is ${{\left( x+2 \right)}^{2}}=9$.
The above equation can be converted into the quadratic equation.
To write the above equation as the quadratic equation. We need to write all the terms on one side and make the equation equal to zero.
Now we get the quadratic equation as.
\[\Rightarrow {{\left( x+2 \right)}^{2}}-9=0\]
The same equation can be written as
$\Rightarrow {{\left( x+2 \right)}^{2}}-{{3}^{2}}=0$
To solve the simplified equation, we can use the algebra identity
$\left[ {{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right) \right]$
Now let’s assume $\left( x+2 \right)$ as $a$and $3$ as $b$.
Now substitute the respective terms in the above algebra identity.
The resultant equation is
$\Rightarrow {{\left( x+2 \right)}^{2}}-{{3}^{2}}=\left( x+2-3 \right)\left( x+2+3 \right)$
Now apply the operations in the bracket to get the simple equation
$\Rightarrow {{\left( x+2 \right)}^{2}}-{{3}^{2}}=\left( x-1 \right)\left( x+5 \right)$
Therefore by solving the equation ${{\left( x+2 \right)}^{2}}=9$ we got, two terms in the resultant equation.
from the resultant terms, we can find the roots of the equation ${{\left( x+2 \right)}^{2}}=9$
let’s take the term $\left( x-1 \right)=0$
$\Rightarrow x-1=0$
$\Rightarrow x=1$
Therefore $x=1$ is one of the root values.
Now let’s take the other term $\left( x+5 \right)=0$
$\Rightarrow \left( x+5 \right)=0$
$\Rightarrow x=-5$
Therefore $x=-5$ is the other root value.
Hence by solving the equation ${{\left( x+2 \right)}^{2}}=9$ , the resultant roots of the $x$ are $1$ and $-5$ .

Note: We know that the given equation is quadratic. Generally, any quadratic equation will have three terms and at least one term with the square exponent. The term which has the square as an exponent will have a coefficient that is not equal to zero.