Question

Class 10

Mathematics

Chapter 4 - Quadratic equations

Formulas Needed

Quality Answer Needed​

Answer 1

⠀⠀⠀⠀⠀★ QUADRATIC EQUATIONS —

❍ When we equates any Quadratic Polynomial with zero then, it becomes a Quadratic equation (ax² + bx + c = 0).

$\underline{\bf{\dag} \:\mathfrak{Quadratic\; Formula\: :}}$⠀⠀⠀⠀

For any Quadratic equation, the roots α and β are given by :

$\bf{\dag}\;\;\underline{\boxed{\frak{\Big(\alpha, \;\beta \Big) = \bigg(\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\bigg)}}}$

where,

• a is coefficient of x²
• b is coefficient of x
• c is constant term
• α and β are roots

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⠀⠀⠀

$\underline{\bf{\dag} \:\mathfrak{Nature\;of\;Roots\: :}}$⠀⠀⠀⠀

• D (Discriminant) = b² - 4ac

If,

• D > 0 (roots are unequal and real)
• D = 0 (roots are real and equal)
• D < 0 (roots are unequal and Imaginary)

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⠀⠀⠀

$\underline{\bf{\dag} \:\mathfrak{Sum\;\&\; Product\;(\alpha\;and \; \beta)\: :}}$⠀⠀⠀⠀

If α and β are roots of any Quadratic equation (ax² + bx + c = 0) then sum and Product is given by :

$\underline{\boxed{\frak{Sum = (\alpha + \beta) = \dfrac{-b}{\;a}\;\&\; Product = (\alpha\;\beta)=\dfrac{c}{a}}}}$

Also,

⠀⠀⠀⠀⠀

For any cubic equation, the roots (α, β and γ) are given by :

$\underline{\boxed{\frak{\alpha + \beta + \gamma = \dfrac{-b}{\;a}, \;\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}, \;\alpha \beta \gamma = \dfrac{-d}{\;a}}}}$

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Answer 2

Quadratic equations :

　　◆ Quadratic formula

• $\sf{{\underline{\boxed{\bf{ {\dfrac{-b \pm {\sqrt{b^{2}-4ac}} }{2a}} }}}}}$

　　◆ Nature of roots

$\sf{b^{2}-4ac}$ is known as Discriminant (D)

• D > 0 (Two distinct real roots)
• D = 0 (Roots are real and equal)
• D < 0 (No real roots)

　　◆ Methods to solve Quadratic

　　　equations :

• By splitting the middle term

For eg :-

$\sf{~~~~~~~~~~ 2x^{2}+4x+2=0}$

$\sf\implies{{\boxed{\sf{\pmb{S=4~,~~P=2×2=4~~~~~~(2,2)}}}}}$

$\sf\implies{(2x^{2}+2x)+(2x+2)=0}$

$\sf\implies{[2x(x+1)~]+[2(x+1)~]=0}$

$\sf\implies{(2x+2)(x+1)=0}$

$\sf\implies{2x+2=0~~~or~~~x+1=0}$

$\sf\implies{x={\dfrac{-2}{2}}~~~and~~~x=-1}$

$\sf\implies{{\blue{\underline{\boxed{\bf{x=-1~~~and~~~x=-1}}}}}}$

$\sf{~~~~~~ \therefore Roots~are:~~-1~,~-1}$

‎

‎

• Using the Quadratic formula (given above)