Question

Using quadratic formula find the solutions of the equation x² +12x + 35 = 0.​

Answer 1

$\large\underline{\bf{Solution-}}$

Given quadratic equation is

$\rm :\longmapsto\: {x}^{2} + 12x + 35 = 0$

$\red{\rm :\longmapsto\:\sf \: On \: comparing \: with \: {ax}^{2} + bx + c \: = 0}$

we get ,

$\red{\rm :\longmapsto\:a = 1 \: \: } \\ \red{\rm :\longmapsto\:b = 12} \\ \red{\rm :\longmapsto\:c = 35}$

Let first find Discriminant, D

$\red{\rm :\longmapsto\:Discriminant, D = {b}^{2} - 4ac}$

$\rm \: \: = \: \: {(12)}^{2} - 4 \times 35 \times 1$

$\rm \: \: = \: \: 144 - 140$

$\rm \: \: = \: \: 40$

$\: \: \: \: \: \: \: \: \: \: \red{\bf\implies \:Discriminant, D = 4}$

Solution using quadratic formula is

$\blue{\bf :\longmapsto\:x = \dfrac{ - b \: \pm \: \sqrt{ D } }{2a} }$

$\rm :\longmapsto\:x = \dfrac{ - 12 \: \pm \: \sqrt{4} }{2 \times 1}$

$\rm :\longmapsto\:x = \dfrac{ - 12 \: \pm \: 2}{2}$

$\rm :\longmapsto\:x = \dfrac{ - 12 \: + \: 2}{2} \: \: or \: \: \dfrac{ - 12 - 2}{2}$

$\rm :\longmapsto\:x = \dfrac{ - 10}{2} \: \: or \: \: \dfrac{ - 14}{2}$

$\bf\implies \:x = - 5 \: \: \: or \: \: \: x = - 7$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac