Question

In the following exercises, solve equations in Quadratic Form for $12x\hspace{0.17em}+\hspace{0.17em}5\sqrt{x}\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}=\hspace{0.17em}0$.

Short answer

The solution of the quadratic equation is $x=\frac{9}{16},x=\frac{1}{9}$.

Step-by-step solution

Step 1. Given Information.

The given equation is $12x\hspace{0.17em}+\hspace{0.17em}5\sqrt{x}\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}=\hspace{0.17em}0$.

Step 2. Remove Square root

Solving $12x\hspace{0.17em}+\hspace{0.17em}5\sqrt{x}\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}=\hspace{0.17em}0$

$$\begin{gathered} {\left(12x-\hspace{0.17em}3\right)}^2={\left(-5\sqrt{x}\right)}^2\hspace{0.17em} \\ 144x^2-72x+9=25x \\ 144x^2-72x-25x+9=0 \\ 144x^2-97x+9=0 \end{gathered}$$

Step 3. Solving using Quadratic equation.

Solving $144x^2-97x+9=0$

$\begin{array}{rcl}{x}_{1,\hspace{0.17em}2}& =& \frac{-\left(-97\right)\pm \sqrt{{\left(-97\right)}^2-4·\hspace{0.17em}144·\hspace{0.17em}9}}{2·\hspace{0.17em}144}\\ x_{1,\hspace{0.17em}2}& =& \frac{-\left(-97\right)\pm \hspace{0.17em}65}{2·\hspace{0.17em}144}\\ x_1& =& \frac{-\left(-97\right)+65}{2·\hspace{0.17em}144},\hspace{0.17em}{x}_2=\frac{-\left(-97\right)-65}{2·\hspace{0.17em}144}\\ x& =& \frac{9}{16},\hspace{0.17em}x=\frac{1}{9}\end{array}$

. Verifying the solution

Plug the value of $x=\frac{9}{16}$in equation $12x\hspace{0.17em}+\hspace{0.17em}5\sqrt{x}\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}=\hspace{0.17em}0$, we get:-

$\begin{array}{rcl}12\left(\frac{9}{16}\right)\hspace{0.17em}+\hspace{0.17em}5\sqrt{\frac{9}{16}}\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}& =& \hspace{0.17em}\frac{27}{4}+\frac{15}{4}-3\\ \frac{27}{4}+\frac{15}{4}-3& =& \frac{42}{4}-3\\ \frac{42}{4}-3& =& \frac{30}{4}\\ \frac{30}{4}& =& \frac{15}{2}\end{array}$

Since the value of $\frac{9}{16}$is the equation satisfied the condition.

Similarly, Plug value of $x=\frac{1}{9}$in equation $12x\hspace{0.17em}+\hspace{0.17em}5\sqrt{x}\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}=\hspace{0.17em}0$, we get:-

localid="1645292125046" $\begin{array}{rcl}12\left(\frac{1}{9}\right)+5\sqrt{\frac{1}{9}}-3& =& \frac{4}{3}+\frac{5}{3}-3\\ \frac{4}{3}+\frac{5}{3}-3& =& \frac{9}{3}-3\\ \frac{9}{3}-3& =& 3-3\\ 3-3& =& 0\end{array}$

Since the value of $x=\frac{1}{9}$the equation satisfied the condition.

Hence, The solution of the equation is localid="1663940752402" $x=\frac{9}{16},x=\frac{1}{9}$.