Question

What is the quadratic equation with roots 3 and 4?

Answer 1

$the \: quadratic \: equation \: with \: roots \: \alpha and \beta \: is \: {x}^{2} + x( \alpha + \beta ) + \alpha \beta = 0 \\ here \: \alpha = 3 \: and \: \beta = 4 \\ hence \: the \: required \: equation \: is \: {x}^{2} + x(3 + 4) + 3 \times 4 = 0 \\ {x}^{2} + 7x + 12 = 0$

Answer 2

The required quadratic equation whose roots are 3 and 4 is x² - 7x + 12 = 0

Given :

A quadratic equation whose roots are 3 and 4

To find :

The quadratic equation with roots 3 and 4

Concept :

If the roots of a quadratic equation is given then the quadratic equation is given by

$\sf{ {x}^{2} -(Sum \: of \: the \: roots )x + Product \: of \: the \: roots }= 0$

Solution :

Step 1 of 2 :

Find Sum of roots and Product of the roots

Here it is given for a quadratic equation whose roots are 3 and 4

Sum of roots = 3 + 4 = 7

Product of the roots = 3 × 4 = 12

Step 2 of 2 :

Find the quadratic equation

The required quadratic equation is given by

$\sf{ {x}^{2} -(Sum \: of \: the \: roots )x + Product \: of \: the \: roots }= 0$

$\displaystyle \sf \implies {x}^{2} - 7x + 12 = 0$

Hence the required quadratic equation whose roots are 3 and 4 is x² - 7x + 12 = 0

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