Question

the quadratic equation 2x²– 8x + p = 0 has two distinct real roots. what is the highest possible integer value of p?​

Answer 1

$\large \dag$ Question :-

The quadratic equation 2x²– 8x + p = 0 has two distinct real roots. What is the highest possible integer value of p ?

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf  \green{Highest   \: Possible \:  Integer }} }\ \\ \green{\underline{\underline{\sf  Value \: of \: p \: is \: 7 }}}$

$\large \dag$ Step by step Explanation :-

We Know that If the roots are real & unequal of any quadratic equation ax² + bx + c then its discriminant is greater than zero i.e.

$\large \bf \red\bigstar \: \: \orange{ \underbrace{ \underline{   \blue{b^2-4ac>0 }}}}$

Here We Have,

2x²– 8x + p = 0 has two distinct real roots

Therefore,

$:\longmapsto \rm {( - 8)}^{2} - 4 \times 2 \times p > 0$

$:\longmapsto \rm 64 - 8p > 0$

$:\longmapsto \rm 8p < 64$

$:\longmapsto \rm p < \cancel \frac{64}{8}$

$\large \red{:\longmapsto} \green{ \underline{ \overline{ \pmb{ \boxed{ \rm p < 8}}}}}$

So largest possible integer value of p less than 8 will be 7 .

Therefore,

$\large\purple\leadsto\large\underline{\pink{\underline{\frak{\pmb{\text Answer \: is \: 7 }}}}}$

$\large \dag$ Additional Information :-

❒ Quadratic Polynomial with one Variable :

✪ The general form of the equation is ax² + bx + c = 0.

Note : 

◆ If a = 0, then the equation becomes to a linear equation.

◆ If b = 0, then the roots of the equation becomes equal but opposite in sign.

◆ If c = 0, then one of the roots is zero. ]

❒ Nature Of Roots :

✪ b² - 4ac is the discriminant of the equation.

Then,

● If b² - 4ac = 0, then the roots are real & equal.

● If b² - 4ac > 0, then the roots are real & unequal.

● If b² - 4ac < 0, then the roots are imaginary.

Answer 2

Step-by-step explanation:

Given :

$\rightarrowtail\bold{2x^{2}-8x+p=0}$

$\rightarrowtail \bold{It\;has\;two\;distinct\;real\;roots.}$

To find :

$\rightarrowtail \bold{Highest\;possible\;integer\;value\;of\;p\;?}$

Solution :

• If discriminant (b² - 4ac) > 0, the quadratic equation has two distinct real roots.

$\rightsquigarrow \bold{2x^2-8x+p=0}$

• b² - 4ac > 0

⇒ (-8)² - 4(2)(p) > 0

⇒ 64 - 8p > 0

⇒ 64/8 - 8p/8 > 0

⇒ 8 - p > 0

⇒ -p > -8

• [When dividing with (-) sign, always inequality reverses]

⇒ p < 8

∴ Largest possible integer value of p is less than 8, that is 7.