Question

The difference of the roots of \[{x^2} - 7x - 9 = 0\] is
A) \[ - 2\]
B) 16
C) \[2\sqrt {85} \]
D) \[\sqrt {85} \]

Answer 1

Hint:
Here, we are required to find the difference of the roots of the given quadratic equation. We will compare the given equation with the general form of quadratic equation. We will then substitute the values in the quadratic formula to get the two roots of the given equation. Subtracting one root from the other will give us the required difference of the roots of the given equation.

Formula Used:
Quadratic Formula: \[x = \dfrac{{ - B \pm \sqrt {{B^2} - 4AC} }}{{2A}}\], where \[A\] is the coefficient of \[{x^2}\], \[B\] is the coefficient of \[x\] and \[C\] is the constant, in the quadratic equation \[A{x^2} + Bx + C = 0\]

Complete Step by Step Solution:
Given quadratic equation is:
\[{x^2} - 7x - 9 = 0\]
Now, comparing this equation with the general form of quadratic equation, \[A{x^2} + Bx + C = 0\]
We can say that,
\[A = 1\], \[B = - 7\] and \[C = - 9\]
Now, in order to find roots of a quadratic equation of the form \[A{x^2} + Bx + C = 0\], we use the quadratic formula \[x = \dfrac{{ - B \pm \sqrt {{B^2} - 4AC} }}{{2A}}\].
Now, for finding the roots of the given quadratic equation \[{x^2} - 7x - 9 = 0\]
We will substitute \[A = 1\], \[B = - 7\] and \[C = - 9\] in the quadratic formula \[x = \dfrac{{ - B \pm \sqrt {{B^2} - 4AC} }}{{2A}}\]. Therefore, we get
\[x = \dfrac{{ - \left( { - 7} \right) \pm \sqrt {{{\left( { - 7} \right)}^2} - 4\left( 1 \right)\left( { - 9} \right)} }}{{2\left( 1 \right)}}\]
Simplifying the expression, we get
\[ \Rightarrow x = \dfrac{{7 \pm \sqrt {49 + 36} }}{2}\]
Adding the terms, we get
\[ \Rightarrow x = \dfrac{{7 \pm \sqrt {85} }}{2}\]
Hence, we can write the two roots as:
\[{x_1} = \dfrac{{7 + \sqrt {85} }}{2}\] and \[{x_2} = \dfrac{{7 - \sqrt {85} }}{2}\]
Now, according to the question, we are required to find the difference of the roots of the given quadratic equation.
Hence, the positive difference of the roots will be given by \[\left| {{x_1} - {x_2}} \right|\].
Substituting \[{x_1} = \dfrac{{7 + \sqrt {85} }}{2}\] and \[{x_2} = \dfrac{{7 - \sqrt {85} }}{2}\] in the expression \[\left| {{x_1} - {x_2}} \right|\], we get
\[\left| {{x_1} - {x_2}} \right| = \left| {\dfrac{{7 + \sqrt {85} }}{2} - \dfrac{{7 - \sqrt {85} }}{2}} \right|\]
\[ \Rightarrow \left| {{x_1} - {x_2}} \right| = \left| {\dfrac{{7 + \sqrt {85} - 7 + \sqrt {85} }}{2}} \right|\]
Subtracting and adding terms, we get
\[ \Rightarrow \left| {{x_1} - {x_2}} \right| = \left| {\dfrac{{2\sqrt {85} }}{2}} \right|\]
Dividing the numerator by 2, we get
\[ \Rightarrow \left| {{x_1} - {x_2}} \right| = \dfrac{{2\sqrt {85} }}{2} = \sqrt {85} \]
Hence, the required difference between the roots of the given quadratic equation \[{x^2} - 7x - 9 = 0\] is \[\sqrt {85} \]

Therefore, option D is the correct answer.

Note:
An alternate way to find the difference between the roots of a quadratic equation is by using formula.
As we have discussed,
The given quadratic equation is: \[{x^2} - 7x - 9 = 0\]
And, after comparing it with the equation \[A{x^2} + Bx + C = 0\], we get,
\[A = 1\], \[B = - 7\] and \[C = - 9\]
Let the roots of the quadratic equation be \[\alpha \] and \[\beta \].
Now, we know that the sum of roots of a quadratic equation \[ = \dfrac{{ - B}}{A}\]
And, the product of roots of a quadratic equation \[ = \dfrac{C}{A}\]
Hence, substituting the values, we get,
\[\left( {\alpha + \beta } \right) = \dfrac{{ - B}}{A} = \dfrac{{ - \left( { - 7} \right)}}{1} = 7\]
And, \[\alpha \cdot \beta = \dfrac{C}{A} = \dfrac{{ - 9}}{1} = - 9\]
Now, we will use the formula:
\[{\left( {\alpha - \beta } \right)^2} = {\left( {\alpha + \beta } \right)^2} - 4\alpha \cdot \beta \]
Here, substituting \[\left( {\alpha + \beta } \right) = 7\] and \[\alpha \cdot \beta = - 9\], we get,
\[{\left( {\alpha - \beta } \right)^2} = {\left( 7 \right)^2} - 4\left( { - 9} \right) = 49 + 36 = 85\]
Taking square root on both sides,
\[ \Rightarrow \left( {\alpha - \beta } \right) = \sqrt {85} \]
Hence, the difference between the roots \[\alpha \] and \[\beta \] of the equation \[{x^2} - 7x - 9 = 0\] is \[\sqrt {85} \]
$\therefore $ Option D is the correct answer.