How do you find the value of the discriminant and determine the nature of the roots $x^2 + 4x = -7$ ?

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Answer 1 · 2016-09-26T09:31:28

Roots are complex conjugates. For given equation they are $x=-2-3i$ or $x=-2+3i$

The discriminant of a quadratic equation $ax^2+bx+c=0$ is $b^2-4ac$ , which decides the nature of roots of the equation.

If $a$ , $b$ and $c$ are rational and $b^2-4ac$ is square of a rational number, roots are rational.

If $b^2-4ac>0$ but is not a square of a rational number, roots are real but not rational.

If $b^2-4ac>0-0$ we have equal roots.

If $b^2-4ac<0$ roots are complex, and if $a$ , $b$ and $c$ are rational. they are complex conjugates

In $x^2+4x=-7hArrx^2+4x+7=0$

the discriminant is $4^2-4xx1xx7=16-28=-12$

hence roots are complex conjugates.

In fact $x^2+4x+7=0$

$hArrx^2+4x+4-(-3)=0$

or $(x+2)^2-(3i^2)=0$

or $(x+2+3i)(x+2-3i)=0$

i.e. $x=-2-3i$ or $x=-2+3i$

How do you find the value of the discriminant and determine the nature of the roots x^2 + 4x = -7x^2 + 4x = -7?