Question

determine the nature of roots for the following quadratic equation 3x²-5x+7=0​

Answer 1

Quadratic Equations

A quadratic equation in a variable $x$ is an equation which is of the form $ax^2 + bx + c = 0$ where constants $a$, $b$ and $c$ are all real numbers and $a \neq 0$.

In case of a quadratic equation $ax^2 + bx + c = 0$ the expression $b^2 - 4ac$ is called the discriminant.

Step-by-step explanation:

We've been given a quadratic equation, and we're asked to find the nature of the roots.

The equation is $3x^2 - 5 + 7$, where;

• a = co-efficient of x² = 3
• b = co-efficient of x = -5
• c = constant term = 7

We know that the nature of the roots of a quadratic equation is given by its discriminant (D).‎‎

Discriminant (D) = b² - 4ac‎‎

Let us consider a quadratic equation $ax^2 + bx + c = 0$, then nature of roots of quadratic equation depends upon Discriminant $D = b^2 - 4ac$ of the quadratic equation.

If $D = b^2 - 4ac > 0$, then roots of the equation are real and unequal.

If $D = b^2 - 4ac = 0$, then roots of the equation are real and equal.

If $D = b^2 - 4ac < 0$, then roots of the equation are unreal or complex or imaginary.

For the quadratic equation $3x^2 - 5 + 7$;

$\implies D = b^2 - 4ac$

$\implies D = (-5)^2 - 4(3)(7)$

$\implies D = 25 - 4(3)(7)$

$\implies D = 25 - 84$

$\implies D = -59$

Here, D = -59, meaning D < 0.

∴ The nature of the roots are imaginary, there are no real roots.

Answer 2

Answer:

The nature of roots of the equation $3x^{2} -5x+7=0$ is complex conjugates.

Step-by-step explanation:

We have the quadratic formula  

$x=\frac{-b}{2a}$±$\frac{\sqrt{b^{2}-4ac } }{2a}$     to find the roots and the discriminant in the quadratic formula $b^{2}-4ac$ determines the nature of roots.

In the given equation $3x^{2} -5x+7=0$,

$a=3,b=-5,c=7$,    substituting in the discriminant we get

$b^{2} -4ac=(-5)^{2} -4(3)(7)$

             $=25-84=-59$ $<0$

since the discriminant is $<0$, we have the conclusion the polynomial has no real roots but it has complex roots.

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