Question

2x^2-13x+9=0 solve by using completing the square method​

Answer 1

Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :

x2-(13/2)x+(9/2) = 0

Subtract 9/2 from both side of the equation :

x2-(13/2)x = -9/2

Now the clever bit: Take the coefficient of x , which is 13/2 , divide by two, giving 13/4 , and finally square it giving 169/16

Add 169/16 to both sides of the equation :

On the right hand side we have :

-9/2 + 169/16 The common denominator of the two fractions is 16 Adding (-72/16)+(169/16) gives 97/16

So adding to both sides we finally get :

x2-(13/2)x+(169/16) = 97/16

Adding 169/16 has completed the left hand side into a perfect square :

x2-(13/2)x+(169/16) =

(x-(13/4)) • (x-(13/4)) =

(x-(13/4))2

Things which are equal to the same thing are also equal to one another. Since

x2-(13/2)x+(169/16) = 97/16 and

x2-(13/2)x+(169/16) = (x-(13/4))2

then, according to the law of transitivity,

(x-(13/4))2 = 97/16

We'll refer to this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(x-(13/4))2 is

(x-(13/4))2/2 =

(x-(13/4))1 =

x-(13/4)

Now, applying the Square Root Principle to Eq. #3.2.1 we get:

x-(13/4) = √ 97/16

Add 13/4 to both sides to obtain:

x = 13/4 + √ 97/16

Since a square root has two values, one positive and the other negative

x2 - (13/2)x + (9/2) = 0

has two solutions:

x = 13/4 + √ 97/16

or

x = 13/4 - √ 97/16

Note that √ 97/16 can be written as

√ 97 / √ 16 which is √ 97 / 4

Answer 2

$2 {x}^{2} - 13x + 9 = 0$

$move \: consonant \: to \: right - hand \: side \: and \: change \: its \: sign$

$2 {x}^{2} - 13x = \red{ - 9}$

$divide \: both \: side \: eq. \: by2$

$\red{ {x}^{2} - \frac{13}{2} x = \frac{ - 9}{2} }$

$add \: ( \frac{13}{4} {)}^{2} to \: both \: side \: eq.$

${x}^{2} - \frac{13x}{2} \red{+ (\frac{13}{4})^{2} } = - \frac{9}{2} \red { + ( \frac{13}{4} {)}^{2} }$

$using \: {a}^{2} - 2ab + {b}^{2} = (a - {b)}^{2} factor \: the \: eq.$

$\red{(x - \frac{13}{4} {)}^{2} } = - 9 + ( \frac{13}{4} {)}^{2}$

${(x - \frac{13}{4} {)}^{2} } = \red{ \frac{97}{16} }$

$solve \: eq. \: for \: x$

$\red{x = \frac{ - \sqrt{97} + 13}{4} }$

$\red{x = \frac{ \sqrt{97} + 13}{4} }$

$the \: eq. \: has \: 2 \: sol.$

$\boxed{ \pink{x = \frac{ - \sqrt{97} + 13}{4} }}$

${\boxed {\orange{{{x = \frac{ \sqrt{97} + 13}{4} }}}}}$