Name: Quadratic Polynomial
Uploaded: 2019-07-19
Duration: 5 min 9 s
Description: When you kick a ball, what path does it take? The ball goes up in the air and then lands on the ground at another point. Quadratic polynomials can explain th...

Question

Solve the equation

7 x - \frac{\sqrt{3 x^{2} - 8 x + 1}}{x} = {(\frac{8}{\sqrt{x}} + \sqrt{x})}^{2}

.

Accepted Answer

Given:

$7 x - \frac{\sqrt{3 x^{2} - 8 x + 1}}{x} = {(\frac{8}{\sqrt{x}} + \sqrt{x})}^{2}$

$\Rightarrow \frac{7 x^{2} - \sqrt{3 x^{2} - 8 x + 1}}{x} = \frac{(8 + x)^{2}}{x}$

$\Rightarrow 7 x^{2} - \sqrt{3 x^{2} - 8 x + 1} = 64 + 16 x + x^{2}$

$\Rightarrow 6 x^{2} - 16 x - \sqrt{3 x^{2} - 8 x + 1} = 64$

Adding $2$ on both sides, we have

$2 (3 x^{2} - 8 x + 1) - \sqrt{3 x^{2} - 8 x + 1} = 66$

Now, let

$\sqrt{3 x^{2} - 8 x + 1} = y$ then we have

$\Rightarrow 2 y^{2} - y = 66$

$\Rightarrow 2 y^{2} - y - 66 = 0$

$\Rightarrow 2 y^{2} - 12 y + 11 y - 66 = 0$

$\Rightarrow 2 y (y - 6) + 11 (y - 6) = 0$

$\Rightarrow (2 y + 11) (y - 6) = 0$

$\Rightarrow y = - \frac{11}{2}, 6$

Now,

$\sqrt{3 x^{2} - 8 x + 1} = y = 6$

$\Rightarrow 3 x^{2} - 8 x - 35 = 0$

$\Rightarrow 3 x^{2} - 15 x + 7 x - 35 = 0$

$\Rightarrow (3 x + 7) (x - 5) = 0$

$∴$ $x = 5, \frac{- 7}{3}$

Also,

$\sqrt{3 x^{2} - 8 x + 1} = y = \frac{- 11}{2}$

$\Rightarrow 3 x^{2} - 8 x + 1 = \frac{121}{4}$

$\Rightarrow 12 x^{2} - 32 x - 117 = 0$

$\Rightarrow x = \frac{32 \pm \sqrt{1024 + 1404 \times 4}}{4 \times 6} = \frac{8 \pm \sqrt{415}}{6}$

Hence, the roots of the given equation are

$x = 5, \frac{- 7}{3}, \frac{8 \pm \sqrt{415}}{6}$ .

Solve the equation $7 x - \frac{\sqrt{3 x^{2} - 8 x + 1}}{x} = {(\frac{8}{\sqrt{x}} + \sqrt{x})}^{2}$ .

Important Questions on Miscellaneous Equations

Learn and Prepare for any exam you want !

Solve the equation 7x-3x2-8x+1x=8x+x2.

Important Questions on Miscellaneous Equations

Learn and Prepare for any exam you want !

Solve the equation $7 x - \frac{\sqrt{3 x^{2} - 8 x + 1}}{x} = {(\frac{8}{\sqrt{x}} + \sqrt{x})}^{2}$ .