Question

2r²+10r+11=0 completing the square

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Answer 1

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

Given quadratic equation is

$\rm :\longmapsto\: {2r}^{2} + 10r + 11 = 0$

can be rewritten as

$\rm :\longmapsto\: {2r}^{2} + 10r = - 11$

Multiply by 2, on both sides, we get

$\rm :\longmapsto\: {4r}^{2} + 20r = - 22$

$\rm :\longmapsto\: {(2r)}^{2} + 2 \times 2r \times 5= - 22$

Adding 25 on both sides, we get

$\rm :\longmapsto\: {(2r)}^{2} + 2 \times 2r \times 5 + 25= - 22 + 25$

$\rm :\longmapsto\: {(2r)}^{2} + 2 \times 2r \times 5 + {5}^{2} = 3$

$\red{\rm :\longmapsto\: {(2r + 5)}^{2} = 3}$

$\rm :\longmapsto\:2r + 5 = \: \pm \: \sqrt{3}$

$\rm :\longmapsto\:2r = \: - \: 5 \: \pm \: \sqrt{3}$

$\rm \implies\:r \: = \: \dfrac{ - 5 \: \pm \: \sqrt{3} }{2}$

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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