Answer
Verified
486.9k+ views
Hint – For solving such a question, use a simple formula of roots of quadratic equation.
Given equation:
$16{x^4} - 20{x^2} + 5 = 0$
Since the power of $x$ is$4\& 2$
So, let${x^2} = t$ in the above equation.
Then the equation becomes:
$16{t^2} - 20t + 5 = 0$
As we know the formula for roots of quadratic equation is:
$\left[ {x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}} \right]$ for any general quadratic equation of the form $a{x^2} + bx + c = 0$
Hence roots of the given quadratic equation are:
$
t = \dfrac{{ - \left( { - 20} \right) \pm \sqrt {{{\left( { - 20} \right)}^2} - \left( {4 \times 16 \times 5} \right)} }}{{2 \times 16}} \\
t = \dfrac{{20 \pm \sqrt {400 - 320} }}{{32}} \\
t = \dfrac{{20 \pm \sqrt {80} }}{{32}} \\
t = \dfrac{{4\left( {5 \pm \sqrt 5 } \right)}}{{32}} \\
t = \dfrac{{\left( {5 \pm \sqrt 5 } \right)}}{8} \\
$
Substituting the value of $x$ in place of $t$ we get:
$
{x^2} = \dfrac{{\left( {5 \pm \sqrt 5 } \right)}}{8} \\
x = \sqrt {\dfrac{{\left( {5 \pm \sqrt 5 } \right)}}{8}} \\
$
Multiplying and dividing numbers inside the root by $2$.
$
x = \sqrt {\dfrac{{2\left( {5 \pm \sqrt 5 } \right)}}{{16}}} \\
x = \dfrac{{\sqrt {10 \pm 2\sqrt 5 } }}{4} \\
$
As we know that
$\sin {36^0} = \dfrac{{\sqrt {10 - 2\sqrt 5 } }}{4}$
Hence, $\sin {36^0}$ is a root of a given quadratic equation.
Note- Whenever you find such type of problems, you can convert your \[4th\] order equation into quadratic equation by assuming some variable as done in the case above, after that with the help of quadratic formula easily evaluate the unknown variable. Formulas of roots of the quadratic equation mentioned above must be remembered in order to solve the quadratic equation easily.
Given equation:
$16{x^4} - 20{x^2} + 5 = 0$
Since the power of $x$ is$4\& 2$
So, let${x^2} = t$ in the above equation.
Then the equation becomes:
$16{t^2} - 20t + 5 = 0$
As we know the formula for roots of quadratic equation is:
$\left[ {x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}} \right]$ for any general quadratic equation of the form $a{x^2} + bx + c = 0$
Hence roots of the given quadratic equation are:
$
t = \dfrac{{ - \left( { - 20} \right) \pm \sqrt {{{\left( { - 20} \right)}^2} - \left( {4 \times 16 \times 5} \right)} }}{{2 \times 16}} \\
t = \dfrac{{20 \pm \sqrt {400 - 320} }}{{32}} \\
t = \dfrac{{20 \pm \sqrt {80} }}{{32}} \\
t = \dfrac{{4\left( {5 \pm \sqrt 5 } \right)}}{{32}} \\
t = \dfrac{{\left( {5 \pm \sqrt 5 } \right)}}{8} \\
$
Substituting the value of $x$ in place of $t$ we get:
$
{x^2} = \dfrac{{\left( {5 \pm \sqrt 5 } \right)}}{8} \\
x = \sqrt {\dfrac{{\left( {5 \pm \sqrt 5 } \right)}}{8}} \\
$
Multiplying and dividing numbers inside the root by $2$.
$
x = \sqrt {\dfrac{{2\left( {5 \pm \sqrt 5 } \right)}}{{16}}} \\
x = \dfrac{{\sqrt {10 \pm 2\sqrt 5 } }}{4} \\
$
As we know that
$\sin {36^0} = \dfrac{{\sqrt {10 - 2\sqrt 5 } }}{4}$
Hence, $\sin {36^0}$ is a root of a given quadratic equation.
Note- Whenever you find such type of problems, you can convert your \[4th\] order equation into quadratic equation by assuming some variable as done in the case above, after that with the help of quadratic formula easily evaluate the unknown variable. Formulas of roots of the quadratic equation mentioned above must be remembered in order to solve the quadratic equation easily.
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Give 10 examples for herbs , shrubs , climbers , creepers