Answer
Verified
482.7k+ 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
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE