Question

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

`sqrt2x^2 + 7x + 5sqrt2 = 0`

Answer 1

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`.