Question

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Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

$\longmapsto$FORMULA TO BE IMPLEMENTED

A general equation of quadratic equation is

$a {x}^{2} + bx + c = 0$

Now one of the way to solve this equation is by SRIDHAR ACHARYYA formula

For any quadratic equation

$a {x}^{2} + bx + c = 0$

The roots are given by

$\displaystyle \: x = \frac{ - b \pm \: \sqrt{ {b}^{2} - 4ac } }{2a}$

$\longmapsto$CALCULATION

The given Quadratic Equation is

${t}^{2} - 2t + 9 = 0$

Comparing with general equation of quadratic equation

$a {t}^{2} + bt + c = 0$

We get

$a = 1 \: , \: b \: = - 2 \: , c = 9$

So by the Sridhar Acharyya formula the roots are given by

$\displaystyle \: t = \frac{ - b \pm \: \sqrt{ {b}^{2} - 4ac } }{2a}$

$\implies \: \displaystyle \: t = \frac{ 2 \pm \: \sqrt{ {( - 2)}^{2} - 4 \times 1 \times 9 } }{2 \times 1}$

$\implies \: \displaystyle \: t = \frac{ 2 \pm \: \sqrt{4 - 36 } }{2}$

$\implies \: \displaystyle \: t = \frac{ 2 \pm \: \sqrt{ - 32 } }{2}$

$\implies \: \displaystyle \: t = \frac{ 2 \pm \: 4i\sqrt{ 2 } }{2}$

$\implies \: \displaystyle \: t = { 1 \pm \: 2i\sqrt{ 2 } }$

RESULT

Hence the required roots are

$\displaystyle \: ( { 1 \ + \: 2i\sqrt{ 2 } } \: ) \: \: \: and \: \: \: ( { 1 \ - \: 2i\sqrt{ 2 } } \: )$

Where i = Imaginary Number

Answer 2

Answer:

According to the Quadratic Formula, t , the solution for At2+Bt+C = 0 , where A, B and C are numbers, often called coefficients, is given by :

- B ± √ B2-4AC

t = ————————

2A

In our case, A = 1

B = -2

C = -9

Accordingly, B2 - 4AC =

4 - (-36) =

40

Applying the quadratic formula :

2 ± √ 40

t = —————

2

Can √ 40 be simplified ?

Yes! The prime factorization of 40 is

2•2•2•5

To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).

√ 40 = √ 2•2•2•5 =

± 2 • √ 10

√ 10 , rounded to 4 decimal digits, is 3.1623

So now we are looking at:

t = ( 2 ± 2 • 3.162 ) / 2

Two real solutions:

t =(2+√40)/2=1+√ 10 = 4.162

or:

t =(2-√40)/2=1-√ 10 = -2.162