Question

$\huge{ \boxed{\underline{\bf{\sf{Question : - }}}}}$
Discuss the nature of the roots of the following equations.
$\sf{ 4{x}^{2} \: - \: 12x \: + \: 9 \: = \: 0}$
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Answer 1

$\huge \fbox \blue{Solution:}$

$\textbf{Given:} \: \sf{ 4{x}^{2} \: - \: 12x \: + \: 9 \: = \: 0}$

$\sf \colorbox{lightgreen} {Here}$

$⇒a = 4$

$⇒b = - 12$

$⇒c = 9$

$⇒ \textbf{D} = ( - 12 {)}^{2} - 4 \times 4 \times 9 = 144 - 144 = 0$

$\text{∴Equation has real and equal roots }$

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$\\ \\ \\ \\ \\ \sf \colorbox{gold} {\red(ANSWER ᵇʸ ⁿᵃʷᵃᵇ⁰⁰⁰⁸}$

Answer 2

$\blue\bf{\bigstar{QuestioN}\bigstar:-}$

Discuss the nature of the roots of the following equations.

$\bf{4x^2-12x+9=0}$

$\pink\bf{\bigstar{AnsweR}\bigstar:-}$

✧To find the nature of roots, we have to use the formula for discriminant.

• $\bf{Discriminant = b^2 - 4ac}$

✧Here,

• $:\longrightarrow\green\bf{a=4~,b=-12~, c=9}$

✧On applying formula,

$:\implies\red\bf{b^2-4ac=(-12)^2-4\times4\times9}$

$:\implies\red\bf{144-4\times4\times9}$

$:\implies\red\bf{144-144}$

${\boxed{\underline{\bf{\sf{\red{\bigstar0\bigstar}}}}}$

✧Here,

✧D = 0

✧That's why  the equation have two equal real roots!!

$\huge{\boxed{\underline{\bf{\sf{\red{\bigstar{Extra~Info:-}\bigstar}}}}}$

⟿ Nature of roots are based on the discriminant.

⟿ According to that, three types are there. They are,

• Two distant real roots.

• Two equal real roots.

• No real roots.

⟿ Nature of roots of quadratic equations by using quadratic formula,

• Two distant real roots if, b² - 4ac > 0.

• Two equal real roots if, b² - 4ac = 0

• No real roots if, b² - 4ac < 0.

Happy Learning Dear!!♡