Question

Solve the quadratic inequality :: 2x^2+x-15>=0

Answer 1

An equation of the form ax2+bx +c = 0 is called a quadratic equation in one variable, where a, b, c are real numbers and a ≠ 0.

Quadratic Formula:

This method is also called as Sridharacharya's rule.


x= –b±√b2 – 4ac/2a

where b2 - 4ac is called the discriminant of the quadratic equation and it is denoted by 'D'.

D= b2 - 4ac

x= -b±√D/2a


Nature of the roots


If D = 0 roots are real and equal , D > 0 roots are real and unequal, D < 0 roots are imaginary.

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Solution:

(i) 2x2 – 7x + 3 = 0

On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = -7 and c = 3

By using quadratic formula, we get
x = –b±√b2 – 4ac/2a

⇒ x = 7±√49 – 24/4
⇒ x = 7±√25/4
⇒ x = 7±5/4
⇒ x = 7+5/4 or x = 7-5/4
⇒ x = 12/4 or 2/4

∴  x = 3 or 1/2

(ii) 2x2 + x – 4 = 0


On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = 1 and c = -4


By using quadratic formula, we get
x = –b±√b2 – 4ac/2a


⇒x = -1±√1+32/4
⇒x = -1±√33/4


∴ x = -1+√33/4 or x = -1-√33/4



(iii) 4x2 + 4√3x + 3 = 0


On comparing this equation with ax2 + bx + c = 0, we get
a = 4, b = 4√3 and c = 3


By using quadratic formula, we get
x = –b±√b2 – 4ac/2a


⇒ x = -4√3±√48-48/8
⇒ x = -4√3±0/8


∴ x = √3/2 or x = -√3/2



(iv) 2x2 + x + 4 = 0


On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = 1 and c = 4


By using quadratic formula, we get
x = –b±√b2 – 4ac/2a


⇒ x = -1±√1-32/4


⇒ x = -1±√-31/4


The square of a number can never be negative.
∴there is no real solution of this equation.



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Hope this will help you.......

Answer 2

Answer:

X = -3, X = 5/2

Step-by-step explanation:

Solving by Quadratic Formula:-

Step 1:- In accordance to the Quadratic Formula,  x  , for the solution ax²+bx+c  = 0  , here  a, b  and  c  are integers, often called coefficients, is given by

X = [-b ± √(b²- 4ac)] / 2a

In our case we have, a = 2, b = 1, c = -15

Step 2:- According to B² - 4AC = 1 - [4 × 2 × (-15 )]

                                                   = 1 - ( - 121 )    

                                                   = 121             [ ( - ) × ( - ) = (+) ]

Step 3:- Applying Quadratic formula:-

                                            X = ( - 1 ± √121 ) / 4

Step 4:- Simplifying √121 = √( 11 × 11 ) = 11

Step 5:- So now,

                          X = ( - 1 ± 11 ) / 4

              we get,

                          X = ( - 1 + √121 ) / 4

                              = 10 / 4         [ ∵ √121 = 11; ( - 1 + 11 ) / 4 ]

                              = 5/2 = 2.500

                          X = ( - 1 - √121 ) / 4

                              = ( - 12 ) / 4 = ( - 3 ).       [ ∵ √121 = 11; ( - 1 - 11 ) / 4 ]

Hence the Answer is X = -3 Or X = 5/2.