Question

How do you solve ${{\left( x+5 \right)}^{2}}=49$?

Answer 1

We first simplify the quadratic equation. We use both grouping methods and vanishing methods to solve the problem. We take common terms out to form the multiplied forms. In the case of the vanishing method, we use the value of x which gives the polynomial value 0.

Complete step-by-step solution:
We have ${{\left( x+5 \right)}^{2}}=49$. Simplifying we get
$\begin{aligned} & {{\left( x+5 \right)}^{2}}=49 \\\ & \Rightarrow {{x}^{2}}+10x+25=49 \\\ & \Rightarrow {{x}^{2}}+10x-24=0 \\\ \end{aligned}$
We apply the middle-term factoring or grouping to factorize the polynomial.
Factorizing a polynomial by grouping is to find the pairs which on taking their common divisor out, give the same remaining number.
In case of ${{x}^{2}}+10x-24$, we break the middle term $10x$ into two parts of $12x$ and $-2x$.
So, ${{x}^{2}}+10x-24={{x}^{2}}+12x-2x-24$. We have one condition to check if the grouping is possible or not. If we order the individual elements of the polynomial according to their power of variables, then the multiple of end terms will be equal to the multiple of middle terms.
Here multiplication for both cases gives $-24{{x}^{2}}$. The grouping will be done for ${{x}^{2}}+12x$ and $-2x-24$.
We try to take the common numbers out.
For ${{x}^{2}}+12x$, we take $x$ and get $x\left( x+12 \right)$.
For $-2x-24$, we take -2 and get $-2\left( x+12 \right)$.
The equation becomes ${{x}^{2}}+10x-24={{x}^{2}}+12x-2x-24=x\left( x+12 \right)-2\left( x+12 \right)$.
Both the terms have $\left( x+12 \right)$ in common. We take that term again and get
$\begin{aligned} & {{x}^{2}}+10x-24 \\\ & =x\left( x+12 \right)-2\left( x+12 \right) \\\ & =\left( x+12 \right)\left( x-2 \right) \\\ \end{aligned}$
Multiple of two polynomials is 0 which means at least one of them has to be 0.
This gives $\left( x+12 \right)\left( x-2 \right)=0$. The roots are $x=2$ or $x=-12$.

Note: We find the value of x for which the function $f\left( x \right)={{x}^{2}}+10x-24=0$. We can see $f\left( 2 \right)={{\left( 2 \right)}^{2}}+10\times 2-24=4+20-24=0$. So, the root of the $f\left( x \right)={{x}^{2}}+10x-24$ will be the function $\left( x-2 \right)$. This means for $x=a$, if $f\left( a \right)=0$ then $\left( x-a \right)$ is a root of $f\left( x \right)$.
Now, $f\left( x \right)={{x}^{2}}+10x-24=\left( x+12 \right)\left( x-2 \right)$. We can also do the same process for $\left( x+12 \right)$.
We also directly solve it where we take root value on both sides and get
$\begin{aligned} & {{\left( x+5 \right)}^{2}}=49 \\\ & \Rightarrow \left( x+5 \right)=\pm \sqrt{49}=\pm 7 \\\ & \Rightarrow x=\left( 7-5 \right)\text{ or }\left( -7-5 \right) \\\ & \Rightarrow x=2\text{ or }\left( -12 \right) \\\ \end{aligned}$