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  • Stuck here, help me understand: Let two numbers have an arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

Let two numbers have an arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

  • Option 1)

    x^{2}+18x-16=0

  • Option 2)

    x^{2}-18x+16=0

  • Option 3)

    x^{2}+18x+16=0

  • Option 4)

    x^{2}-18x-16=0

 

Answers (1)

As we learnt in 

Sum of Roots in Quadratic Equation -

\alpha +\beta = \frac{-b}{a}

- wherein

\alpha \: and\beta are root of quadratic equation

ax^{2}+bx+c=0

a,b,c\in C

 

 

Product of Roots in Quadratic Equation -

\alpha \beta = \frac{c}{a}

- wherein

\alpha \: and\ \beta are roots of quadratic equation:

ax^{2}+bx+c=0

a,b,c\in C

 

 

Geometric mean of two numbers (GM) -

GM= \sqrt{ab}

- wherein

It is to be noted that a,G,b are in GP and a,b are two non - zero numbers.

 

\\Let\ two\ numbers\ be\ \alpha ,\beta \\*\\*\frac{\alpha +\beta }{2}=9= > \alpha +\beta =18\\*\\*\sqrt{\alpha \beta }=4= > \alpha \beta =16

Thus quadratic equation is x^{2}-18x+16=0


Option 1)

x^{2}+18x-16=0

Incorrect

Option 2)

x^{2}-18x+16=0

Correct

Option 3)

x^{2}+18x+16=0

Incorrect

Option 4)

x^{2}-18x-16=0

Incorrect

Posted by

Vakul

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