Question

if the quadratic equation x^2+tx+25=0 has two equal real roots find the value of t​

Answer 1

Question :-

The quadratic equation x² + tx + 25 = 0 has two equal real roots. Find the value of 't'.

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

Given that,

$\rm \: {x}^{2} + tx + 25 = 0 \: has \: real \: and \: equal \: roots.$

We know,

A quadratic equation ax² + bx + c = 0 has real roots iff Discriminant, D = b² - 4ac = 0

So, on comparing the given equation with ax² + bx + c = 0,

we have

$\rm \: a = 1$

$\rm \: b = t$

$\rm \: c = 25$

So, as equation have equal roots.

$\rm\implies \:Discriminant, D = 0$

$\rm \: {b}^{2} \: - \: 4ac \: = \: 0$

$\rm \: {t}^{2} \: - \: 4 \times 1 \times 25 \: = \: 0$

$\rm \: {t}^{2} \: - \: 100 \: = \: 0$

$\rm \: {t}^{2} \: = \: 100$

$\rm\implies \: \: t \: = \: \pm \: 10$

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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

Answer 2

$\color{lime}\large \bold{Given,}$

$\sf {x}^{2} \: + \: tx \: + \: 25 \: = \: 0 \: \: \: \: \color{aqua} { \underline{Has \: real \: and \: equal \: roots.}}$

$\sf \: a \: = \: 1 \\ \\ \sf \: b \: = \: t \\ \\ \sf \: c \: = \: 25$

$\sf \: We\:know\:that,\:A\: quadratic\: equation \: {a}^{2} \: + \: bx \: + \: c \: = \: 0 \: has \: real \: roots \: and \: \\ \sf Discriminate ,\: D \: = \: {b}^{2} \: - \: 4ac \: = \: 0 \\ \\ \sf So \: lets \: compare \: the \: given \: equation \: ―$

$\sf \: We\:know\:that,\:A\: quadratic\: equation \: {a}^{2} \: + \: bx \: + \: c \: = \: 0 \: has \: real \: roots \: and \: \\ \sf Discriminate ,\: D \: = \: {b}^{2} \: - \: 4ac \: = \: 0 \\ \\ \sf So \: lets \: compare \: the \: given \: equation \: with \: a {x}^{2} \: + \: bx \: + \: c \: = \: 0$

$\sf \: Discriminate\:=\:0 \\ \\ \sf \: b^2\:-\:4ac\:=\:0 \\ \\ \sf \: {t}^{2} \: - \: 4 \: \times \: 1 \: \times \: 25 \: = \: 0 \\ \\ \sf \: {t}^{2} \: - \: 100 \: = \: 0 \\ \\ \sf \: {t}^{2} \: = \: 100 \\ \\ \sf \: t \: = \: \sqrt{100} \\ \\ \sf \: t \: = \: ± \: 10$