Question

Find the roots of following equations:
(1) \[x-\dfrac{1}{x}=3,x\ne 0\]
(2) \[\dfrac{1}{x+4}-\dfrac{1}{x-7}=\dfrac{11}{30},x\ne -4,7\]

Answer 1

Hint: First we have to find the value of discriminant \[{{b}^{2}}-4ac\] and check if the roots are real and unequal or not. Then substituting the values of a, b, c in formula \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] to find the roots of the equation.

Complete step by step answer:
(1) We know the general quadratic equation is given as \[a{{x}^{2}}+bx+c=0\] where a, b are the coefficients of \[{{x}^{2}}\] and x respectively and c is the constant in the equation.
By expressing the given equation \[x-\dfrac{1}{x}=3\] in the form of a general quadratic equation we get,
Multiplying with x on both sides of the equation,
\[{{x}^{2}}-1=3x\]
By subtracting with 3x on both sides we get,
\[{{x}^{2}}-3x-1=0\]
Comparing the above equation with the general quadratic equation we get the values of a, b, c as,
\[a=1\], \[b=-3\], \[c=-1\]
We know the formula to find the roots of a general quadratic equation is given as \[\dfrac{-b\pm \sqrt{D}}{2a}\] where \[D={{b}^{2}}-4ac\] is known as the discriminant.
Applying the above formula by substituting the values of a, b, c to find the discriminant,
For the roots to be real and unequal the discriminant must be greater than 0.
\[\left[ {{\left( -3 \right)}^{2}}-4\left( 1 \right)\left( -1 \right) \right]>0\]
Squaring \[-3\] and multiplying \[-4\] with\[-1\] we get,
\[\left( 9+4 \right)>0\]
Adding 9 with 4 the resultant is greater than 0,
13 > 0
Hence the roots are real and unequal.
Applying the formula to find the roots of a general quadratic equation by substituting the values of a, b, c and the value of the discriminant \[{{b}^{2}}-4ac\] we get,
\[\dfrac{-\left( -3 \right)\pm \sqrt{13}}{2}\]
By taking square root of 13 and simplifying the expression for \[\dfrac{-b+\sqrt{D}}{2a}\] and \[\dfrac{-b-\sqrt{D}}{2a}\] we get,
\[\begin{align}
  & x=3.3027 \\
 & x=-0.3027 \\
\end{align}\]
Hence the roots of the equation \[x-\dfrac{1}{x}=3\] are 3.3027 and -0.3027.
(2)
By expressing the given equation \[\dfrac{1}{x+4}-\dfrac{1}{x-7}=\dfrac{11}{30}\] in the form of a general quadratic equation we get,
Taking a common denominator \[\left( x+4 \right)\left( x-7 \right)\] in the LHS,
\[\dfrac{\left( x-7 \right)-\left( x+4 \right)}{\left( x+4 \right)\left( x-7 \right)}=\dfrac{11}{30}\]
Simplifying the above equation,
\[\dfrac{x-7-x-4}{\left( x+4 \right)\left( x-7 \right)}=\dfrac{11}{30}\]
\[\dfrac{-11}{{{x}^{2}}-7x+4x-28}=\dfrac{11}{30}\]
Cross multiplying the terms to express as a quadratic equation,
Cancelling out 11 and multiplying with \[-1\] on both sides,
\[{{x}^{2}}-3x-28=-30\]
Expressing the equation in the simplest form possible,
\[{{x}^{2}}-3x+2=0\]
Comparing the above equation with the general quadratic equation we get the values of a, b, c as,
\[a=1\], \[b=-3\], \[c=2\]
We know the formula to find the roots of a general quadratic equation is given as \[\dfrac{-b\pm \sqrt{D}}{2a}\] where \[D={{b}^{2}}-4ac\] is known as the discriminant.
Applying the above formula by substituting the values of a, b, c to find the discriminant,
For the roots to be real and unequal the discriminant must be greater than 0.
\[\left[ {{\left( -3 \right)}^{2}}-4\left( 1 \right)\left( 2 \right) \right]>0\]
Squaring \[-3\] and multiplying \[-4\] with 2 we get,
\[\left( 9-8 \right)>0\]
Subtracting 8 from 9 the resultant is greater than 0,
1 > 0
Hence the roots are real and unequal.
Applying the formula to find the roots of a general quadratic equation by substituting the values of a, b, c and the value of the discriminant \[{{b}^{2}}-4ac\] we get,
\[\dfrac{-\left( -3 \right)\pm \sqrt{1}}{2}\]
By taking square root of 1 and simplifying the expression for \[\dfrac{-b+\sqrt{D}}{2a}\] and \[\dfrac{-b-\sqrt{D}}{2a}\] we get,
x = 2
x = 1
Hence the roots of the equation \[\dfrac{1}{x+4}-\dfrac{1}{x-7}=\dfrac{11}{30}\] are 2 and 1.

So, the correct answer is “Option A”.

Note: The discriminant plays a major role which specifies the nature of roots therefore calculate the discriminant D carefully. The roots can be either real or imaginary which are determined by discriminant. If the value of discriminant is less than 0 then no real roots exist and the roots are imaginary.