Question

QUESTION 1 PART 1 AND 2 EXPLAIN

Answer 1

Roots of the equation are -1 and 1.8

so a)-2.24 b)6.832

Answer 2

$\bold{\blue{\underline{\red{G}\pink{iv}\green{en}\purple{:-}}}}$

• f(x) = 5x² + 4x - 9
• 'α' and 'β' are the zeros of the polynomial .

$\bold{\blue{\underline{\red{Zeros}\pink{\:of\:}\green{\:the\:polynomials}\purple{:-}}}}$

• Zeros of the polynomial is the value of the variable for which when replaced in the polynomial the value for the polynomial changes to zero.

$\bold{\blue{\underline{\red{General}\pink{\:form \:of }\green{\:polynomial}\purple{:-}}}}$

• $\bold{a_{n}x^n + a_{n-1}x^{n-1} +...... + a_{2}x^2 + a_{1}x + a_{0}}$

• Where ' $\bold{a_{n} , a_{n-1} ...... a_{1} , a_{0}}$ ' are Real numbers and 'n' is an integer .

$\bold{\blue{\underline{\red{Q}\pink{uest}\green{ion}\purple{:-}}}}$

To find:

i) α² - β²

ii) α³ + β³

$\bold{\blue{\underline{\red{S}\pink{olut}\green{ion}\purple{:-}}}}$

    $\bold{\blue{\underline{\red{To}\pink{\:find\:}\green{\alpha\:and\:\beta}\purple{:}}}}$

• 5x² + 4x - 9
• mn = -45  
• m + n = 4x
• m = 9 & n = -5
• 5x² + 9x - 5x - 9       {Splitting the middle term}
• 5x(x - 1) + 9(x - 1)
• (5x + 9)(x - 1)
• 5x + 9 = 0

• x = $\bold{\frac{-9}{5}}$

       $\bold{\boxed{\green{\alpha\:=\:\frac{-9}{5}}}}$

• x - 1 = 0
• x = 1

       $\bold{\boxed{\green{\beta\:=\:1}}}$

     $\bold{\blue{\underline{\red{To}\pink{\:find\:}\green{the\:solution}\purple{:}}}}$

• Replace the values of 'α' and 'β' in the equations:

   α² - β²:

• α² - β² = $\bold{{\frac{-9}{5^2}}}^{2}$ - 1²

• α² - β² = $\bold{\frac{81}{25}}$ - 1

• α² - β² = $\bold{\frac{81 \:- \:25}{25}}$

• α² - β² = $\bold{\frac{56}{25} }$

       $\bold{\boxed{\green{\alpha ^{2}\:-\:\beta ^{2}\:=\:2.24 }}}$    

    α³ + β³:

• α³ + β³ = $\bold{{\frac{-9}{5^3}}}^{3}$ + 1³

• α³ + β³ = $\bold{{\frac{-729}{125}}}$ + 1

• α³ + β³ = $\bold{{\frac{-729\:-\:125}{125}}}$

• α³ + β³ = $\bold{{\frac{-854}{125}}}$

        $\bold{\boxed{\green{\alpha ^{3}\:+\:\beta ^{3}\:=\:-6.832 }}}$