Question

Determine the nature of the roots for the quadrats
equotion.
3x
${?}^{2}$
+2x-2=0​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

$\large\underline{\bold{Given \:Question - }}$

$\sf \: Determine \: the \: nature \: of \: the \: roots \: for \: the \: quadratic \: equation :$

$\sf \: {3x}^{2} + 2x - 2 = 0$

$\large\underline{\bold{Solution-}}$

Concept Used :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac

Here,

The given quadratic equation is

$\sf \: {3x}^{2} + 2x - 2 = 0$

On comparing with ax² + bx + c = 0, we get

$\: \: \: \: \: \: \: \: \bull \: \sf \: a \: = \: 3$

$\: \: \: \: \: \: \: \: \bull \: \sf \: b \: = \: 2$

$\: \: \: \: \: \: \: \: \bull \: \sf \: c \: = \: - \: 2$

Now,

Discriminant, of quadratic equation is given by

$\bf \: Discriminant, D \: = \sf \: {b}^{2} - 4ac$

On substituting the values of a, b and c, we get

$\bf \: Discriminant, D \: = \sf \: {(2)}^{2} - 4 \times (3) \times ( - 2)$

$\bf \: Discriminant, D \: = \sf \: 4 + 24$

$\bf \: Discriminant, D \: = \sf \: 28$

$\rm :\implies\:\bf \: Discriminant, D \: > \: 0$

$\bf \: Hence, \: equation \: has \: real \: and \: distinct \: roots.$

Additional Information :-

A quadratic equation is an equation of degree 2, mean that the highest exponent of this equation is 2. Moreover, the standard quadratic equation is ax² + bx + c, where a, b, and c are real numbers or arbitrary constants and ‘a’ cannot be 0. An example of quadratic equation is 2x² + 5x + 4.

The quadratic expression can also be written as:

• Standard Form: y = ax² + bx + c, here a, b, and c are real numbers.

• Factored Form: y = (ax + c) (bx + d) here also a, b, and c are real numbers.

• Vertex Form: y = a (x + b)² + c, and here also a, b, and c are real numbers.