Question

Write the nature of roots of quadratic equation 4x8-5x+2=0.*
O (a) no real roots
O (b) equal roots
O (c) real and unequal
0 (d) unequal
O Other
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Answer 1

Question : -

Write the nature of roots of a quadratic equation 4x² - 5x + 2 = 0 .

• ( a ) No real roots

• ( b ) Equal roots

• ( c ) Real and unequal

• ( d ) unequal

• ( e ) other

Answer : -

Given : -

Quadratic equation ;

4x² - 5x + 2 = 0

Required to find : -

• Nature of the roots ?

Concept used : -

Before solving this question we need to know some concept which can be used to solve this question .

We know that ;

The quadratic formula is ,

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

Here ,

b² - 4ac is known to be as " discriminate " .

It is denoted by letter ' D ' .

The reason behind naming this as discriminate is ;

This b² - 4ac will be discriminating the nature of the roots .

From the value of b² - 4ac we can conclude the nature of the roots .

The conditions are ;

• If D = 0

The roots are real , equal & rational .

• If D > 0

The roots are real , unequal , rational ( If the result is a perfect square ) .

• If D > 0

The roots are real , unequal , irrational ( If the result is not a perfect square ).

• If D < 0

The roots are imaginary , unequal .

Solution : -

Quadratic equation

4x² - 5x + 2 = 0

we need to find the nature of the roots .

So,

The standard form of the quadratic equation is ;

• ax² + bx + c

Now,

Let's compare the standard form of the quadratic equation with the given quadratic equation .

$\overline{ \star \red{\downarrow} \: \: \: \: \: \: \: \blue{ \downarrow}\: \: \: \: \: \: \: \purple{ \downarrow } \: \: \: \: \red{ \downarrow }\: \: \: \: \: \: \: \: \: \ \blue{ \downarrow }\: \: \: \: \: \: \: \: \: \purple{ \downarrow} \: } \\ \mathtt{a {x}^{2} + bx + c \: \: \: \: \: 4 {x}^{2} - 5x + 2}$

Hence,

• a = 4

• b = - 5

• c = 2

The discriminate is ;

D = b² - 4ac

Substituting the values in the above one ;

=> D = ( - 5 )² - 4 ( 4 ) ( 2 )

=> D = 25 - 32

=> D = - 7

Since,

• D < 0

The roots are imaginary , unequal . ( no real roots )

Therefore,

Option - a is correct ✓

We don't have any real roots for the given quadratic equation .

Answer 2

$\huge\bf\underline\red{SoluTiOn\: : \: -}$

Qᴜᴀᴅʀᴀᴛɪᴄ ᴇQᴜᴀᴛɪᴏɴ

Qᴜᴀᴅʀᴀᴛɪᴄ ᴇQᴜᴀᴛɪᴏɴ 4x² - 5x + 2 = 0

ᴡᴇ ɴᴇᴇᴅ ᴛᴏ ꜰɪɴᴅ ᴛʜᴇ ɴᴀᴛᴜʀᴇ ᴏꜰ ᴛʜᴇ ʀᴏᴏᴛꜱ .

ꜱᴏ,

ꜱᴏ, ᴛʜᴇ ꜱᴛᴀɴᴅᴀʀᴅ ꜰᴏʀᴍ ᴏꜰ ᴛʜᴇ Qᴜᴀᴅʀᴀᴛɪᴄ ᴇQᴜᴀᴛɪᴏɴ ɪꜱ ;

ᴀx² + ʙx + ᴄ

ɴᴏᴡ, ʟᴇᴛ'ꜱ ᴄᴏᴍᴘᴀʀᴇ ᴛʜᴇ ꜱᴛᴀɴᴅᴀʀᴅ ꜰᴏʀᴍ ᴏꜰ ᴛʜᴇ Qᴜᴀᴅʀᴀᴛɪᴄ ᴇQᴜᴀᴛɪᴏɴ ᴡɪᴛʜ ᴛʜᴇ ɢɪᴠᴇɴ Qᴜᴀᴅʀᴀᴛɪᴄ ᴇQᴜᴀᴛɪᴏɴ .

$\begin{lgathered}\overline{ \star \red{\downarrow} \: \: \: \: \: \: \: \orange{ \downarrow}\: \: \: \: \: \: \: \red{ \downarrow } \: \: \: \: \orange{ \downarrow }\: \: \: \: \: \: \: \: \: \ \red{ \downarrow }\: \: \: \: \: \: \: \: \: \orange{ \downarrow} \: } \\ \mathtt{a {x}^{2} + bx + c \: \: \: \: \: 4 {x}^{2} - 5x + 2}\end{lgathered}$

Hence,

• a = 4
• b = - 5
• c = 2

The discriminate is ;

D = b² - 4ac

Substituting the values in the above one ;

$\rightarrow$D = ( - 5 )² - 4 ( 4 ) ( 2 )

$\rightarrow$D = 25 - 32

$\rightarrow$ D = - 7

Since,

ᴅ < 0

ᴛʜᴇ ʀᴏᴏᴛꜱ ᴀʀᴇ ɪᴍᴀɢɪɴᴀʀʏ , ᴜɴᴇQᴜᴀʟ . ( ɴᴏ ʀᴇᴀʟ ʀᴏᴏᴛꜱ )

ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴏᴘᴛɪᴏɴ - ᴀ ɪꜱ ᴄᴏʀʀᴇᴄᴛ ✓ ᴡᴇ ᴅᴏɴ'ᴛ ʜᴀᴠᴇ ᴀɴʏ ʀᴇᴀʟ ʀᴏᴏᴛꜱ ꜰᴏʀ ᴛʜᴇ ɢɪᴠᴇɴ Qᴜᴀᴅʀᴀᴛɪᴄ ᴇQᴜᴀᴛɪᴏɴ .