x2 + 5x – 14 = 0, x = √2 , –7, 3

(iii) x² + 5x – 14 = 0, x = √2 , –7, 3

Sol. a) By putting x =  √2 in L.H.S. we get

L.H.S. = ( √2 )² + 5( √2 ) – 14

= 2 + 5√ 2 – 14

= 5 √2 – 12

≠ R.H.S.

∴  L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So,  √2 is not the root of the given quadratic equation.

b) By putting x = –7 in L.H.S. we get

L.H.S. = (–7)² + 5(–7) – 14

= 49 – 35 – 14

= 49 – 49

= 0

= R.H.S.

∴ L.H.S. = R.H.S.

Thus equation is satisfied.

So,  –7 is the root of the given quadratic equation.

c) By putting x = 3 in L.H.S. we get

L.H.S. = (3)² + 5(3) – 14

= 9 + 15 – 14

= 24 – 14

= 10

≠ R.H.S.

∴ L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So,  3 is not the root of the given quadratic equation.

4m2 – 9 = 0, m = 2, 2/3, 3/2

(ii) 4m² – 9 = 0, m = 2, 2/3, 3/2

Sol. a) By putting m = 2 in L.H.S. we get

L.H.S. = 4(2)² – 9

= 4(4) – 9

= 16 – 9

= 7

≠ R.H.S.

∴  L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So, 2 is not the root of the given quadratic equation.

b) By putting m = 2/3 in L.H.S. we get

L.H.S. = 4(2/3)² – 9

= 4(4/9) – 9

= (16/9 )- 9

= (16 – 81)/9

= - 65 / 9

≠ RHS

∴  L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So, 2/3 is not the root of the given quadratic equation.

c) By putting m = 3/2 in L.H.S. we get

L.H.S. = 4(3/2)² – 9

= 4(9/4) – 9

= 9 – 9

= 0

= R.H.S.

∴  L.H.S. = R.H.S.

Thus equation is satisfied.

So, 3/2 is the root of the given quadratic equation.

x2 + 3x – 4 = 0, x = 1, –2, – 3

(i) x² + 3x – 4 = 0,  x = 1, –2, – 3

Sol. a) By putting x = 1 in L.H.S. we get

L.H.S. = (1)² + 3(1) – 4

= 1 + 3 – 4

= 4 – 4

= 0

= R.H.S.

∴  L.H.S. = R.H.S.

Thus equation is satisfied.

So, 1 is the root of the given quadratic equation.

b) By putting x = –2 in L.H.S. we get

L.H.S. = (–2)² + 3(–2) – 4

= 4 – 6 – 4

= – 6

≠ R.H.S.

∴  L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So,  –2 is not the root of the given quadratic equation.

c) By putting x = –3 in L.H.S. we get

L.H.S. = (–3)² + 3(–3) – 4

= 9 – 9 – 4

= – 4

≠ R.H.S.

∴  L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So,  –3 is not the root of the given quadratic equation.

Quadratic equations

Definition

A quadratic equation in the variable 'x' is an equation of the form ax² + bx + c = 0

where a, b, c  are real numbers and a is not equal to zero. 

Nature of roots of a quadratic equation

Relation between roots and coefficients of a quadratic equation

Formation of quadratic equation when roots are given

A car left 30 minutes later than the scheduled time. in order to reach its destination 150 km away in time, it has to increase its speed by 25 km/hr from its usual speed. Find its usual speed.

The base of a triangle is 4 cm longer than its altitude. If the area of the triangle is 48 sq. cm, then find its base and altitude.

The sum of a number and its reciprocal is 5 1/5 . Find the numbers.

Solve by factorization method : 6x2 – 5x – 25 = 0

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0 , find the value of (α+1/β) (1/α+β)

If α and β are the toots of the equation 2x^2 - 3x - 1 = 0, find the values of α^2/β+β^2/α

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of α - β.

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of (α/β) + (β/α)

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of a^2 + b^2

If the sum and product of the roots of the quadratic equation ax^2 - 5x + c = 0 are both equal to 10, then find the values of a and c.

If one of the roots of the equation 3x^2 - 10x + k = 0 is 1/3, then find the other root and also the value of k.

Find the values of k so that the equation x^2 - 2x ( 1 + 3k) + 7 (3 + 2x) = 0 has real and equal roots.

Determine the nature of roots of the following quadratic equation 2x^2 + 5x + 5 = 0 .

Determine the nature of roots of the following quadratic equation 4x^2 - 28x + 49 = 0

Determine the nature of roots of the following quadratic equation x^2 - 11x - 10 = 0

If α and β are the roots of the equation 3x^2 - 4x + 1 = 0. Form a quadratic equation whose roots are α^2 / β and β^2 /α

Form the quadratic equation whose roots are 7 + √3 and 7 - √3.

Try these Questions Yourself 

Find the values of k for which the roots are real and equal in each of the following equations.

Determine the nature of the roots of real equation. (Try your self)

If α and β are the roots of the equation 3x^2 - 5x + 2 = 0, then find the values of

Form a quadratic equation whose roots are

Find the sum and product of the roots of the following equations.

QUADRATIC EQUATION IMPORTANT SUMS FOR PRACTICE

Chapter No. 2. Quadratic Equation, SSC Maths I, Maharashtra

Practice Set 2.1

Practice Set 2.2

Practice Set 2.3

Practice Set 2.4

Practice Set 2.5

Practice Set 2.6

Problem Set No. 2