Question

If the equation \[\left( {a + 1} \right){x^2} - \left( {a + 2} \right)x + \left( {a + 3} \right) = 0\] has roots equal in magnitude but opposite in sign. Then what are the roots of the equation?
A. \[ \pm a\]
B. \[ \pm \left( {\dfrac{1}{2}} \right)a\]
C. \[ \pm \left( {\dfrac{3}{2}} \right)a\]
D. \[ \pm 2a\]

Answer 1

Hint: First, by using the coefficients of a quadratic equation and find the sum and product of the roots of the given quadratic equation. Since roots are equal in magnitude and opposite in signs. So, equate the sum of the roots of the equation to 0. Simplify the equation and find the value of \[a\]. Then substitute the value of \[a\] in the given quadratic equation to get the required answer

Formula Used:
If \[p\] and \[q\] are the roots of the equation \[a{x^2} + bx + c = 0\], then
Sum of the roots: \[p + q = \dfrac{{ - b}}{a}\]
Product of the roots: \[pq = \dfrac{c}{a}\]

Complete step by step solution:
The given quadratic equation is \[\left( {a + 1} \right){x^2} - \left( {a + 2} \right)x + \left( {a + 3} \right) = 0\].
The roots of the equation are equal in magnitude and opposite in signs.
We know that, if the roots of a quadratic equation are equal in magnitude and opposite in signs, then the sum of the roots is zero.
The sum of the roots of the given quadratic equation is,
\[sum = - \dfrac{{ - \left( {a + 2} \right)}}{{\left( {a + 1} \right)}}\]
\[ \Rightarrow \]\[sum = \dfrac{{\left( {a + 2} \right)}}{{\left( {a + 1} \right)}}\]
Now equate the sum with zero.
\[\dfrac{{\left( {a + 2} \right)}}{{\left( {a + 1} \right)}} = 0\]
\[ \Rightarrow \]\[a + 2 = 0\]
\[ \Rightarrow \]\[a = - 2\] \[.....\left( 1 \right)\]

Now substitute the value of \[a\] in the given quadratic equation.
\[\left( { - 2 + 1} \right){x^2} - \left( { - 2 + 2} \right)x + \left( { - 2 + 3} \right) = 0\]
Simplify the above equation.
\[ - {x^2} - \left( 0 \right)x + 1 = 0\]
\[ \Rightarrow \]\[{x^2} - 1 = 0\]
\[ \Rightarrow \]\[{x^2} = 1\]
Take square root on both sides.
\[x = \pm 1\] \[.....\left( 2 \right)\]

When substituting the value of \[a\] in the given options.
The option \[ \pm \left( {\dfrac{1}{2}} \right)a\] satisfies the equations \[\left( 1 \right)\] and \[\left( 2 \right)\].
Thus, \[ \pm \left( {\dfrac{1}{2}} \right)a\] are the roots of the given quadratic equation.

Hence the correct option is B.

Note: Students often get confused about when to use the quadratic formula and when to use the sum and product of the roots.
If it is asked to calculate the roots of a quadratic equation, then we use the quadratic formula.
If it is asked to calculate the roots of a quadratic equation with some conditions of the roots, then we use the sum and product of the quadratic equation.