Question

find the roots of quadratic equation 16x^2 -24x - 1= 0 by using quadratic formula

Answer 1

Answer:

The roots are $x=\frac{3+\sqrt{10}}{4},\frac{3-\sqrt{10}}{4}$

Step-by-step explanation:

Given : The quadratic equation - $16x^2 -24x - 1= 0$

To find : The roots of quadratic equation ?

Solution :

Using quadratic formula,

General form - $ax^2+bx+c=0$

$D=b^2-4ac$

Solution is $x=\frac{-b\pm\sqrt{D}}{2a}$

Equation is $16x^2 -24x - 1= 0$

where, a=16 , b=-24, c=-71

$D=b^2-4ac$

$D=(-24)^2-4(16)(-1)$

$D=576+64$

$D=640$

Solution is $x=\frac{-b\pm\sqrt{D}}{2a}$

$x=\frac{-(-24)\pm\sqrt{640}}{2(16)}$

$x=\frac{24\pm8\sqrt{10}}{16}$

$x=\frac{3\pm\sqrt{10}}{4}$

Therefore, The roots are $x=\frac{3+\sqrt{10}}{4},\frac{3-\sqrt{10}}{4}$

Answer 2

Given:

A quadratic equation 16x² - 24x - 1 = 0.

To Find:

The roots of the above quadratic equation using the quadratic formula.

Solution:

The given problem can be solved using the concepts of quadratic equations.

1. Consider a quadratic equation ax² + b x + c = 0. The roots of the quadratic equation are given by the formula,

=> Roots = $(\frac{-b+\sqrt{b^2 - 4ac} }{2a }, \frac{-b-\sqrt{b^2 - 4ac} }{2a } )$.

2. Use the formula mentioned above to find the roots.

=> 16x^2 -24x - 1= 0, ( a = 16, b = -24 ),

=> Roots = $(\frac{24 + \sqrt{(24)^2 - 4 * 16*(-1)} }{2*16} ,\frac{24 - \sqrt{(24)^2 - 4 * 16*(-1)} }{2*16})$,

=> Roots = $(\frac{24+\sqrt{640} }{32}, \frac{24-\sqrt{640} }{32} )$,

=> Roots = $(\frac{8*3+8\sqrt{10} }{8*4} ,\frac{8*3-8\sqrt{10} }{8*4})$,

=> Roots = [(3+√10)/4, (3-√10)/4].

3. Hence the roots of the equation are (3±√10)/4.

Therefore, the roots of the equation 16x² - 24x - 1 = 0 are$( \frac{3+\sqrt{10} }{4}, \frac{3-\sqrt{10} }{4})$.