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# In the Following, Determine Whether the Given Quadratic Equation Have Real Roots and If So, Find the Roots: Sqrt2x^2+7x+5sqrt2=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:

sqrt2x^2+7x+5sqrt2=0

#### Solution

We have been given, sqrt2x^2+7x+5sqrt2=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=sqrt2, b = 7 and c=5sqrt2.

Therefore, the discriminant is given as,

D=(7)^2-4(sqrt2)(5sqrt2)

= 49 - 40

= 9

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=(-(7)+-sqrt9)/(2(sqrt2))

=(-7+-3)/(2sqrt2)

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

x=(-7+3)/(2sqrt2)

=-sqrt2

Also,

x=(-7-3)/(2sqrt2)

=-5/sqrt2

Therefore, the roots of the equation are -5/sqrt2 and -sqrt2

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In the Following, Determine Whether the Given Quadratic Equation Have Real Roots and If So, Find the Roots: Sqrt2x^2+7x+5sqrt2=0 Concept: Relationship Between Discriminant and Nature of Roots.