Question

form the quadratic equation, if roots are 2/3 and 3/2​

Answer 1

Answer:

Required quadratic equation $6 {x}^{2} - 13x + 6 = 0$

Step-by-step explanation:

Given,a quadratic equation has two roots

$\frac{2}{3}$ and $\frac{3}{2}$

We can write,

$(x - \frac{2}{3} )(x - \frac{3}{2} ) = 0$

${x}^{2} - \frac{3}{2} x - \frac{2}{3} x + \frac{2}{3} \times \frac{3}{2} = 0 \\ {x}^{2} - ( \frac{3}{2} + \frac{2}{3} )x + 1 = 0 \\ {x}^{2} - (\frac{9 + 4}{6} ) x+ 1 = 0 \\ {x}^{2} - \frac{13}{6} x + 1 = 0 \\ 6 {x}^{2} - 13x + 6 = 0$

This is a problem of Algebra.

Some important Algebra formulas.

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab − b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)³ − 3ab(a + b)

a³ - b³ = (a -b)³ + 3ab(a - b)

a² − b² = (a + b)(a − b)

a² + b² = (a + b)² − 2ab

a² + b² = (a − b)² + 2ab

a³ − b³ = (a − b)(a² + ab + b²)

a³ + b³ = (a + b)(a² − ab + b²)

Know more about Algebra,

1) https://brainly.in/question/13024124

2) https://brainly.in/question/1169549

Answer 2

Answer:

The quadratic equation with roots  $\frac{2}{3}$ and $\frac{3}{2}$ is given by 6x² - 13x + 6= 0

Step-by-step explanation:

Given,

The roots of the quadratic equation are $\frac{2}{3}$ and $\frac{3}{2}$

To find,

The quadratic equation

Solution:

Recall the concepts

The formula of the quadratic equation with roots α and β is given by

x² - (sum of roots)x + product of roots = 0

x² - (α+β)x + αβ = 0 ----------------(1)

Since the roots are  $\frac{2}{3}$ and $\frac{3}{2}$, we have α=  $\frac{2}{3}$ and β =  $\frac{3}{2}$

Then, sum of roots = α+β =    $\frac{2}{3}$ + $\frac{3}{2}$ = $\frac{13}{6}$

Product of roots = αβ = are  $\frac{2}{3}$ × $\frac{3}{2}$ = 1

Hence the required quadratic equation is  

x² - ($\frac{13}{6}$)x + 1 = 0

Multiplying throughout by 6 we get

6x² - 13x + 6= 0

∴ The quadratic equation with roots  $\frac{2}{3}$ and $\frac{3}{2}$ is given by 6x² - 13x + 6= 0

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