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In the Following, Determine Whether the Given Quadratic Equation Have Real Roots and If So, Find the Roots: Sqrt3x^2+10x-8sqrt3=0 - Mathematics

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Question

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

sqrt3x^2+10x-8sqrt3=0

Solution

We have been given, sqrt3x^2+10x-8sqrt3=0

Now we also know that for an equation ax2 + bx + c = 0, the discriminant is given by the following equation:

D = b2 - 4ac

Now, according to the equation given to us, we have,a=sqrt3, b = 10 and c=-8sqrt3.

Therefore, the discriminant is given as,

D=(10)^2-4(sqrt3)(-8sqrt3)

= 100 + 96

= 196

Since, in order for a quadratic equation to have real roots, D ≥ 0.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

x=(-b+-sqrtD)/(2a)

Therefore, the roots of the equation are given as follows,

x=(-(10)+-sqrt196)/(2(sqrt3))

=(-10+-14)/(2sqrt3)

=(-5+-7)/sqrt3

Now we solve both cases for the two values of x. So, we have,

x=(-5+7)/sqrt3

=2/sqrt3

Also,

x=(-5-7)/sqrt3

=-4sqrt3

Therefore, the roots of the equation are 2/sqrt3 and -4sqrt3.

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In the Following, Determine Whether the Given Quadratic Equation Have Real Roots and If So, Find the Roots: Sqrt3x^2+10x-8sqrt3=0 Concept: Relationship Between Discriminant and Nature of Roots.