Question

How do you find the discriminant , describe the number and type of root, and find the exact solution using the quadratic formula given \[{x^2} + 4x + 3 = 0\] ?

Answer 1

Hint: In this question, we have to solve the given equation.
First we need to find out the discriminant. Then analysing the discriminant we can evaluate that it is greater or equal or less than zero accordingly we can state that the roots are unequal real roots or equal roots or complex roots. After solving this equation by Sridharacharya’s formula we will get the required solution.

Formula used: The roots of the quadratic equation \[a{x^2} + bx + c = 0\] is
 \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step-by-step solution:
It is given that, \[{x^2} + 4x + 3 = 0 \ldots \ldots .\left( 1 \right)\]
We need to first find out the discriminant.
The discriminant of the quadratic equation \[a{x^2} + bx + c = 0\] is
 \[\Delta = {b^2} - 4ac\]
If \[\Delta > 0\] , then the quadratic equation has two unequal real roots.
If \[\Delta = 0\] , then the quadratic equation has two equal real roots.
If \[\Delta < 0\] , then the quadratic equation has complex roots.
Here in the given equation \[{x^2} + 4x + 3 = 0\] , the discriminant is
 \[\Delta = {4^2} - \left( {4 \times 1 \times 3} \right) = 16 - 12 = 4 > 0\]
Since the discriminant is greater than zero thus, the quadratic equation has two unequal real roots.
Therefore solving the equation \[{x^2} + 4x + 3 = 0\] by Sridharacharya’s formula, we get,
 $ \Rightarrow $ \[x = \dfrac{{ - 4 \pm \sqrt {{4^2} - \left( {4 \times 1 \times 3} \right)} }}{{2 \times 1}}\]
On squaring and multiply the terms and we get
 $ \Rightarrow $ \[x = \dfrac{{ - 4 \pm \sqrt {16 - \left( {12} \right)} }}{{2 \times 1}}\]
Let us subtract the term and we get
 $ \Rightarrow $ \[x = \dfrac{{ - 4 \pm \sqrt 4 }}{2}\]
On taking square root and we get
 $ \Rightarrow $ \[x = \dfrac{{ - 4 \pm 2}}{2}\]
Let us split the term and further we simplify we get,
 $ \Rightarrow $ \[{x_1} = \dfrac{{ - 4 + 2}}{2} = - 1\] and \[{x_2} = \dfrac{{ - 4 - 2}}{2} = - 3\]

Hence we get, the value of x is either \[ - 1\] or \[ - 3\]

Note: If \[a = 0\] then it will become a linear equation not quadratic as there is no \[a{x^2}\] term.
The discriminant is the part of the quadratic formula under the square root.
 \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
 The discriminant of the quadratic equation \[a{x^2} + bx + c = 0\] is
 \[\Delta = {b^2} - 4ac\]
A positive discriminant indicates that the quadratic equation has two unequal real roots.
A discriminant of zero indicates that the quadratic equation has repeated real roots.
A negative discriminant indicates that the quadratic equation has complex roots.
Quadratic equation: In algebra a quadratic equation is any equation that can be rearranged in standard form as \[a{x^2} + bx + c = 0\] , where x represents an unknown and a, b, c represent known numbers where a ≠ 0.