Question

Find the roots (if they exist) of the following quadratic equations by the method of completing the square:
x^2–2 √5 x+1 = 0​

Answer 1

Step-by-step explanation:

x^2 - 2✓5 x + 1 =0

✓D = ✓ 20 - 4

= ✓16

= 4

alpha = (-(-2✓5) + 4 ) / (2*1)

= (2✓5 + 4)/2

= 2 + ✓5

beta = (-(-2✓5) - 4 ) / (2*1)

= (2✓5 - 4)/2

= ✓5 - 2

Answer 2

Answer

a). 2x2−7x+3=0

⇒x2−27x=−23

Adding (47)2 on both sides

⇒x2−27x+(47)2=2−3+(47)2

⇒(x−47)2=2−3+1649

⇒(x−47)2=1625

⇒(x−47)2=(45)2

Taking square root on both sides

⇒(x−47)=±45

⇒x−47=45,     x−47=4−5

     x=45+47      x=4−5+47

     x=3       x=21

b). 2x2+x−4=0

      x2+2x=2

Adding (41)2 on both sides

⇒x2+2x+(41)2=2+(41)2

     (x+41)2=2+161

     (x+41)2=1633

Taking square root on both sides 

⇒x+41=±433

⇒x=±433−41,     x=4−33−41

⇒x=±433−1,     x=4−33−1

c). 4x2+432+3=0

⇒x2+3x+43=0

     x2+3x=4−3

Adding (23)62 on both sides

⇒x2+3x+(23)2=4−3+(23)2

⇒(x+

⇒(x+23)2=0

     x=2−3,2−3

same roots.

d). 2x2+x+4=0

⇒x2+2x+2=0

     x2+2x=−2

Adding (41)2 on both sides

⇒x2+2x+(41)2=−2+(41)62

⇒(x+41)2=−2+161

⇒(x+41)2=−16−31

Hence, solved.