Question

Solve for $x;\,\sqrt{6x+7}-\left( 2x-7 \right)=0$

Answer 1

Hint: Here we have been given an equation in one variable and we have to find the value of that unknown variable. Firstly as we can see that the one value is in square root form so we will take that value on the other side and square both the sides. Then we will simplify the value using algebraic identity. Finally we will use the quadratic formula and get our desired answer.

Complete step-by-step answer:
We have to solve the equation below:
$\,\sqrt{6x+7}-\left( 2x-7 \right)=0$
Now we will take the non square root term on the right side of the equal side as follows:
$\sqrt{6x+7}=\left( 2x-7 \right)$
On squaring both side we get,
$\Rightarrow {{\left( \sqrt{6x+7} \right)}^{2}}={{\left( 2x-7 \right)}^{2}}$
$\Rightarrow 6x+7={{\left( 2x-7 \right)}^{2}}$
Now use the algebraic identity ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ above where $a=2x$ and $b=7$ we get,
$\Rightarrow 6x+7={{\left( 2x \right)}^{2}}-2\times 2x\times 7+{{7}^{2}}$
$\Rightarrow 6x+7=4{{x}^{2}}-28x+49$
Bring all terms on one side as follows:
$\Rightarrow 4{{x}^{2}}-28x+49-6x-7=0$
$\Rightarrow 4{{x}^{2}}-34x+42=0$…..$\left( 1 \right)$
The quadratic formula for any equation of the form
$a{{x}^{2}}+bx+c=0$ …..$\left( 2 \right)$
Where $a,b,c$ are any constant is given as,
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$…..$\left( 3 \right)$
Comparing equation (1) and (2) we get,
$a=4,b=-34,c=42$
On substituting the value in equation (3) we get,
$\Rightarrow x=\dfrac{-\left( -34 \right)\pm \sqrt{{{\left( -34 \right)}^{2}}-4\times 4\times 42}}{2\times 4}$
$\Rightarrow x=\dfrac{34\pm \sqrt{1156-672}}{8}$
Solving further we get,
$\Rightarrow x=\dfrac{34\pm \sqrt{484}}{8}$
Now as we know $484$ is perfect square of $22$ so,
$\Rightarrow x=\dfrac{34\pm 22}{8}$
So we get the two values as,
$\Rightarrow x=\dfrac{34+22}{8}$
$\Rightarrow x=7$
And,
$\Rightarrow x=\dfrac{34-22}{8}$
$\Rightarrow x=\dfrac{12}{8}$
Simplifying further,
$\Rightarrow x=\dfrac{3}{2}$
So we get the values as $x=7,\dfrac{3}{2}$
Hence on solving $\sqrt{6x+7}-\left( 2x-7 \right)=0$ we get the value of $x$ as $7,\dfrac{3}{2}$ .
So, the correct answer is “$7,\dfrac{3}{2}$”.

Note: Algebraic equations are the equations having the variable and the constant along with the algebraic operations whereas algebraic identities are the equations that are true for all the values for the variable in it. In this question we have first separated the square root value by the non-square root value so that when we square them it gets easy to simplify the values. We can use the middle term method instead of quadratic formula and will get the same answer.