Question

Determine the nature of the roots 4x 2 –20x+25=0 whose roots are .

Answer 1

Answer:

${ax}^{2} + bx + c = 0 \\ \\ nature \: of \: roots \: is \: decided \: by \: the \: discriminate \: \\ d = {b}^{2} - 4ac \\if \: a \: b \: c \: are \: r.n \: and \: \: d \: is \: the \: sq. \: of \: r.n \: \\ roots \: = rational \: \\ d > 0 \: but \: not \: sq. \: r.n \: \\ roots \: = real \: but \: not \: rational \: \\ d = 0 \: its \: a \: equal \: roots \: \\ d < 0 \: complex \: roots \: \\ {4x}^{2} - 20x + 25 = 0 \\ a = 4 \\ b = - 20 \\ c = 25 \\ \\ d = {b}^{2} - 4ac \\ d = ( { - 20)}^{2} - 4 \times 4 \times 25 \\ d = 400 - 16 \times 25 \\ d = 400 - 400 \\ d = 0 \\ \\ i t \: has \: equal \: roots \: \\ \\ hope \: its \: helpful \: to \: you \: mate \: .$

Answer 2

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Solution :-

Given Equation,

= 4x² - 20x + 25 = 0

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

Here, a = 4 , b = -20 , and c = 25

We know that,

Discriminant, D = b² - 4ac

D = b² - 4ac

⇒ D = (- 20)² - 4 × 4 × 25

⇒ D = 400 - 400

⇒ D = 0

Since, b² - 4ac = 0

Hence, the given equation has real and equal roots.

More About the Topic :-

For the quadratic equation ax² + bx + c = 0, the expression b² - 4ac is known as discriminant i.e Discriminant, D = b² - 4ac

Nature of roots of a quadratic equation:-

(i). If b² - 4ac > 0, the quadratic equation has two distinct real roots.

(ii). If b² - 4ac = 0, the quadratic equation has two equal real roots.

(iii). If b² - 4ac < 0, the quadratic equation has no real roots.

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