Question

find the root of x minus 1 by X + 2 + X - 3 by x minus 2 equal to 11 by 8 by using quadratic formula​

Answer 1

Step-by-step explanation:

Given that:find the root of x minus one upon X + 2 + x minus 3 upon x minus 2 is equal to 11 upon 8 by using quadratic formula

Expression :

$\bold{\frac{x - 1}{x + 2} + \frac{x - 3}{x - 2} = \frac{11}{8}}$

Solution:

To find the roots of given expression by quadratic equation,first we have to simply the expression and convert it into quadratic equation

Steps for simplification

1.Take LCM

$\frac{(x - 1)(x - 2) + (x - 3)(x + 2)}{ {x}^{2} - 4} = \frac{11}{8}$

2.Multiply terms in bracket and put common terms together

$\frac{ {x}^{2} - 2x - x + 2 + {x}^{2} + 2x - 3x - 6 }{ {x}^{2} - 4} = \frac{11}{8} \\ \\ \frac{2 {x}^{2} - 4x - 4 }{ {x}^{2} - 4 } = \frac{11}{8}$

3. Cross multiply

$8(2 {x}^{2} - 4x - 4) = 11( {x}^{2} - 4) \\ \\ 16 {x}^{2} - 32x - 32 = 11 {x}^{2} - 44 \\ \\ 5 {x}^{2} - 32x + 12 = 0$

Now apply quadratic formula

$a {x}^{2} + bx + c = 0 \\ \\\bold{ x_{1,2} = \frac{ - b ± \sqrt{ {b}^{2} - 4ac } }{2a} }$

On compare with standard equation

here

$a = 5 \\ b = - 32 \\ c = 12 \\ \\ x_{1,2} = \frac{ - ( - 32) ± \sqrt{ {( - 32)}^{2} - 4 \times 5 \times 12}}{5 \times 2} \\ \\x_{1,2} = \frac{32 ± \sqrt{1024 - 240} }{10} \\ \\ x_{1,2} = \frac{32 ± \sqrt{784} }{10} \\ \\ x_{1,2} = \frac{32± 28}{10}$

So,to find both the roots,put negative and positive value of 28

$x_1 = \frac{32 + 28}{10} \\ \\ x_1 = \frac{60}{10} \\ \\ \bold{x_1 = 6 }\\ \\ x _2= \frac{32 - 28}{10} \\ \\ x_2 = \frac{4}{10 } \\ \\ \bold{x_2= \frac{2}{5}}$

Thus,roots of given expression are 6 and 2/5.

Hope it helps you.