Which equation is quadratic in form?
$3 x^5+8 x^3+6=0$
$6 x^4+7 x^2-3=0$
$5 x^6+x^4+12=0$
$x^9+x^3-10=0$
Answer (1)
Check each equation to see if it can be written in the form a u 2 + b u + c = 0 .
For 6 x 4 + 7 x 2 − 3 = 0 , substitute u = x 2 to get 6 u 2 + 7 u − 3 = 0 , which is quadratic in u .
The other equations cannot be written in this form.
Therefore, the equation 6 x 4 + 7 x 2 − 3 = 0 is quadratic in form: 6 x 4 + 7 x 2 − 3 = 0 .
Explanation
Understanding Quadratic Form We need to determine which of the given equations can be written in the form a u 2 + b u + c = 0 , where u is a function of x . This form is known as quadratic in form.
Checking Each Equation Let's examine each equation:
3 x 5 + 8 x 3 + 6 = 0 : If we let u = x 3 , then u 2 = x 6 , not x 5 . So, this equation is not quadratic in form.
6 x 4 + 7 x 2 − 3 = 0 : If we let u = x 2 , then u 2 = x 4 . Substituting, we get 6 u 2 + 7 u − 3 = 0 . This is a quadratic equation in u .
5 x 6 + x 4 + 12 = 0 : If we let u = x 4 , then u 2 = x 8 , not x 6 . So, this equation is not quadratic in form.
x 9 + x 3 − 10 = 0 : If we let u = x 3 , then u 2 = x 6 , not x 9 . So, this equation is not quadratic in form.
Identifying the Quadratic Form Equation From the above analysis, only the equation 6 x 4 + 7 x 2 − 3 = 0 can be written in the quadratic form 6 u 2 + 7 u − 3 = 0 by substituting u = x 2 .
Final Answer Therefore, the equation that is quadratic in form is 6 x 4 + 7 x 2 − 3 = 0 .
Examples
Quadratic form equations are useful in various fields, such as physics and engineering, where they can simplify complex equations. For example, in analyzing the motion of a damped harmonic oscillator, the equation might involve terms with x 4 and x 2 . By recognizing the quadratic form, we can solve for x 2 and then find the values of x , making the problem more manageable. This technique allows engineers and physicists to solve problems that would otherwise be very difficult.
