Question

solve the quadratic equation
1/x+1/x+3=7/10​

Answer 1

Answer is

x =( 1+√839 i )/2

x =( 1-√839 i )/2

Answer 2

Question : -

Solve this using the Quadratic formula;

1/x + 1/x + 3 = 7/10

Answer : -

Given : -

1/x + 1/x + 3 = 7/10

Required to find : -

• Value ?

Formula used : -

$\boxed{\tt{ x = \dfrac{ - b \pm \sqrt{ b^2 - 4ac}}{2a}}}$

Solution : -

1/x + 1/x + 3 = 7/10

Taking LCM

➙ x + 3 + x / x ( x + 3 ) = 7 / 10

➙ 2x + 3/ x² + 3x = 7/10

➙ By cross multiplication ;

➙ 10 ( 2x + 3 ) = 7 ( x² + 3x )

➙ 20x + 30 = 7x² + 21x

➙ 0 = 7x² + 21x - 30 - 20x

➙ 7x² + 21x - 20x - 30 = 0

➙ 7x² + x - 30 = 0

Hence,

Quadratic equation = 7x² + x - 30

Now,

Let's solve this quadratic equation !

The standard form of a quadratic equation is

ax² + bx + c = 0

Compare this standard form with the Quadratic Equation .

Here,

• a = 7

• b = 1

• c = - 30

Using the Quadratic formula ;

$\boxed{\tt{ x = \dfrac{ - b \pm \sqrt{ b^2 - 4ac}}{2a}}}$

$\tt{ x = \dfrac{ - ( 1 ) \pm \sqrt{ 1^2 - 4 ( 7 ) ( - 30 ) }}{2 ( 7 ) }} \\ \\ \tt{x = \dfrac{ - 1 \pm \sqrt{1 - 4 ( - 210 ) }}{14}} \\ \\ \tt{x = \frac{ - 1 \pm \sqrt{1 - ( - 840)} }{14} } \\ \\ \tt{x = \frac{ - 1 \pm \sqrt{1 + 840} }{14} } \\ \\ \tt{x = \frac{ - 1 \pm \sqrt{841} }{14} }$

This implies ;

$\tt x = \dfrac{ - 1 + \sqrt{841} }{14} \quad ( \: and \: ) \quad x = \dfrac{ - 1 - \sqrt{841} }{14} \\ \\ \tt x = \frac{ - 1 + 29}{14} \quad ( \: and \: ) \quad x = \frac{ - 1 - 29}{14} \\ \\ \tt x = \frac{28}{14} \quad ( \: and \: ) \quad x = \frac{ - 30}{14} \\ \\ \tt x = 2 \quad ( \: and \: ) \quad x = \frac{ - 15}{7}$

Therefore,

7x² + x - 30 = 2 & - 15/7

Additional Information : -

$\boxed{\tt{ x = \dfrac{ - b \pm \sqrt{ b^2 - 4ac}}{2a}}}$

Here,

b² - 4ac is called to be as Discriminate .

This is denoted by "D".

The reason behind this is this b² - 4ac wil be able to discriminate the nature of the roots .

what is nature of roots ?

The nature of roots is the property i.e whether they are rational , irrational , imaginary etc .

So,

The conditions are ;

• If D > 0 ( if it's a perfect square )

The roots are rational , distinct

• If D > 0 ( if it's a non - perfect square )

The roots are irrational , distinct

• If D = 0

The roots are equal and rational

• If D < 0

The roots are distinct and imaginary .

Moreover,

Any example of an imaginary is ;

√-1 = i ( iota )

The imaginary numbers belongs to complex numerical system .

To solve an quadratic equation we can use 3 methods ;

1st method ;

• Factorisation

2nd method ;

• By completing the square

3rd method ;

• Using the quadratic formula

Mostly using the Quadratic formula helps us to solve the complex quadratic equations .