# Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula

**Solution:**

If the given quadratic equation is ax^{2} + bx + c = 0, then:

If b^{2} - 4ac ≥ 0, then the roots are x = [-b ±√(b^{2} - 4ac)]/2a

If b^{2} - 4ac < 0, then no real roots exist.

(i) 2x^{2} - 7x + 3 = 0

a = 2, b = -7, c = 3

b* ^{2}* - 4ac = (-7)

^{2}- 4(2)(3)

= 49 - 24 = 25

b* ^{2}* - 4ac > 0

∴ Roots are x = [-b ±√(b^{2} - 4ac)]/2a

x = [-(- 7) ±√(25)]/2(2)

x = (7 ± 5)/4

x = (7 + 5)/4 and x = (7 - 5)/4

x = 12/4 and x = 2/4

x = 3 and x = 1/2

Roots are 3, 1/2

(ii) 2x^{2} + x - 4 = 0

a = 2, b = 1, c = -4

b^{2} - 4ac = (1)^{2} - 4(2)(-4)

= 1 + 32 = 33

b^{2} - 4ac > 0

∴ Roots are x = [-b ±√(b^{2} - 4ac)]/2a

= [- 1 ± √(33)]/2(2)

= (- 1 ± √(33)/4

x = = [-1 + √(33)]/4 and x = [- 1 - √(33)]/4

Roots are: [- 1 + √(33)]/4, [- 1 - √(33)]/4

(iii) 4x^{2} + 4√3 x + 3 = 0

a = 4, b = (4√3), c = 3

b^{2} - 4ac = (4√3)^{2} - 4(4)(3)

= 48 - 48 = 0

b^{2} - 4ac = 0

∴ Roots are x = [-b ±√(b^{2} - 4ac)]/2a

= [-b ± 0]/2a

= -b/2a

= - 4√3/2(4)

= - √3/2

Roots are - √3/2, - √3/2

(iv) 2x^{2} + x + 4 = 0

a = 2, b = 1, c = 4

b^{2} - 4ac = (1)^{2} - 4(2)(4)

= 1- 32 = - 31

b^{2} - 4ac < 0

∴ No real roots exist.

**Video Solution:**

## Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula

### Class 10 Maths NCERT Solutions - Chapter 4 Exercise 4.3 Question 2:

Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula

The roots of the quadratic equation by applying the quadratic formula are (i) 3, 1/2, (ii) [- 1 + √(33)]/4, [- 1 - √(33)]/4, (iii) - √3/2, - √3/2, (iv) no real roots