Full solution

Q. Solve for

x

2(\sqrt{9x^2+x-7})+5=3x

Isolate square root term: Step $1$ : Simplify the equation by isolating the square root term. $\newline$ Original equation: $2(\sqrt{9x^2+x-7})+5=3x$ $\newline$ Subtract $5$ from both sides: $\newline$ $2(\sqrt{9x^2+x-7}) = 3x - 5$
Divide to isolate square root: Step $2$ : Divide both sides by $2$ to further isolate the square root. $\newline$ $\frac{2(\sqrt{9x^2+x-7})}{2} = \frac{3x - 5}{2}$ $\newline$ $\sqrt{9x^2+x-7} = \frac{3x - 5}{2}$
Eliminate square root: Step $3$ : Square both sides to eliminate the square root. $\newline$ $(\sqrt{9x^2+x-7})^2 = \left(\frac{3x - 5}{2}\right)^2$ $\newline$ $9x^2 + x - 7 = \frac{(3x - 5)^2}{4}$
Expand and simplify: Step $4$ : Expand and simplify the right side of the equation. $\newline$ $9x^2 + x - 7 = \frac{9x^2 - 30x + 25}{4}$ $\newline$ Multiply everything by $4$ to clear the fraction: $\newline$ $4(9x^2 + x - 7) = 9x^2 - 30x + 25$ $\newline$ $36x^2 + 4x - 28 = 9x^2 - 30x + 25$
Form quadratic equation: Step $5$ : Bring all terms to one side to form a quadratic equation. $\newline$ $36x^2 + 4x - 28 - 9x^2 + 30x - 25 = 0$ $\newline$ $27x^2 + 34x - 53 = 0$
Use quadratic formula: Step $6$ : Use the quadratic formula to solve for $x$ . $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 27$ , $b = 34$ , $c = -53$ $x = \frac{-34 \pm \sqrt{34^2 - 4\cdot27\cdot(-53)}}{2\cdot27}$ $x = \frac{-34 \pm \sqrt{1156 + 5724}}{54}$ $x = \frac{-34 \pm \sqrt{6880}}{54}$
Calculate discriminant: Step $7$ : Calculate the discriminant and simplify. $\newline$ $\sqrt{6880} \approx 82.96$ $\newline$ $x = \frac{-34 \pm 82.96}{54}$ $\newline$ $x_1 = \frac{48.96}{54} \approx 0.907$ $\newline$ $x_2 = \frac{-116.96}{54} \approx -2.166$