Question

Solve: $\sqrt {2x + 9} + x = 13$
A. $8$
B. $20$
C. $16$
D. $3$

Answer 1

Hint:
Here we will find the value of $x$ by using the concept of square of numbers. Firstly we will take the square root term separately from all the other terms on one side. Then we will square both sides and get all the terms removed from the square root. Finally we will solve the equation by using Quadratic Formula and get our desired answer.

Formula Used:
Quadratic Formula for equation: $a{x^2} + bx + c = 0$
$x = \dfrac{{\left( { - b} \right) \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step by step solution:
The equation given to us is,
$\sqrt {2x + 9} + x = 13$
So, we will take the$x$variable with no square root from left side to right side as follows,
$\sqrt {2x + 9} = 13 - x$
Now, we will square both side and get,
${\left( {\sqrt {2x + 9} } \right)^2} = {\left( {13 - x} \right)^2}$
$ \Rightarrow {\left( {2x + 9} \right)^{\dfrac{1}{2} \times 2}} = {\left( {13 - x} \right)^2} \\
   \Rightarrow 2x + 9 = {\left( {13 - x} \right)^2} \\$
Next, we will use ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ formula on right side as,
$ 2x + 9 = {\left( {13} \right)^2} - 2 \times 13 \times x + {x^2} \\
   \Rightarrow 2x + 9 = 169 - 26x + {x^2} \\ $
Now we will take all the terms on one side and get,
${x^2} - 26x + 169 - 2x - 9 = 0$
$ \Rightarrow {x^2} - 28x + 160 = 0$…….$\left( 1 \right)$
As, we know the formula to solve the equation in form of $a{x^2} + bx + c = 0$ is,
$x = \dfrac{{\left( { - b} \right) \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Using the above formula in equation (1) we get,
$a = 1,b = - 28,c = 160$
So,
$x = \dfrac{{\left( { - \left( { - 28} \right)} \right) \pm \sqrt {{{\left( { - 28} \right)}^2} - 4 \times 1 \times 160} }}{{2 \times 1}} \\
   \Rightarrow x = \dfrac{{28 \pm \sqrt {784 - - 640} }}{2} \\
   \Rightarrow x = \dfrac{{28 \pm \sqrt {144} }}{2} \\ $
Now we know square root of 144 is 12 so we get,
$x = \dfrac{{28 \pm 12}}{2} \\
   \Rightarrow x = 14 \pm 6 \\ $
On adding the term once and subtracting the term secondly we get,
$x = 14 + 6 = 20 \\
  x = 14 - 6 = 8 \\$
So we got two values of $x$ as 20 and 8.

Hence, option (A) and (B) is correct.

Note:
We can solve the equation by completing the square and rewrite the part of the equation as a perfect square trinomial. A quadratic equation always has two roots; they can be real roots or imaginary roots depending on the values of$a,b,c$. There are various ways to solve a quadratic equation such as Factoring by inspection, Completing the square and Geometric Interpretation.