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2 (i) Express
sqrt(18 x+27) in the form
ksqrt(2x+3), where
k is an integer.
(ii) Hence solve the equation
sqrt(18 x+27)-sqrt(2x+3)=6.

$2$ (i) Express $\sqrt{18 x+27}$ in the form $k \sqrt{2 x+3}$ , where $k$ is an integer. $\newline$ (ii) Hence solve the equation $\sqrt{18 x+27}-\sqrt{2 x+3}=6$ .

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Transformations of linear functions

Full solution

Q.

2

(i) Express

\sqrt{18 x+27}

in the form

k \sqrt{2 x+3}

, where

k

is an integer.

\newline

(ii) Hence solve the equation

\sqrt{18 x+27}-\sqrt{2 x+3}=6

.

Identify common factor: Identify the common factor in $18x+27$ to simplify the square root. $\newline$ $\sqrt{18x+27} = \sqrt{9\cdot 2x + 9\cdot 3}$
Factor out common factor: Factor out the common factor from the square root. $\sqrt{9\cdot 2x + 9\cdot 3} = \sqrt{9}\cdot\sqrt{2x+3}$
Calculate square root: Calculate the square root of the common factor. $\sqrt{9} = 3$
Write simplified form: Write the simplified form of $\sqrt{18x+27}$ . $\sqrt{18x+27} = 3\sqrt{2x+3}$
Solve using simplified form: Now, solve the equation using the simplified form. $3\sqrt{2x+3} - \sqrt{2x+3} = 6$
Combine like terms: Combine like terms on the left side of the equation. $\newline$ $(3-1)\sqrt{2x+3} = 6$
Simplify left side: Simplify the left side of the equation. $2\sqrt{2x+3} = 6$
Divide to isolate square root: Divide both sides of the equation by $2$ to isolate the square root. $\sqrt{2x+3} = 3$
Square both sides: Square both sides of the equation to eliminate the square root. $\newline$ $(\sqrt{2x+3})^2 = 3^2$
Subtract to solve for $x$ : Simplify both sides of the equation. $2x+3 = 9$
Calculate $x$ : Subtract $3$ from both sides of the equation to solve for $x$ . $2x = 9 - 3$
Calculate $x$ : Subtract $3$ from both sides of the equation to solve for $x$ .
$2x = 9 - 3$ Calculate the value of $x$ .
$2x = 6$
$x = \frac{6}{2}$
$x = 3$

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