Question

solve the quadratic equations x²+x-6=0 by any methods

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Answer 1

Step-by-step explanation:

Solution :-

Factorization (Splitting the middle term)Method :-

Given equation is x²+x-6 = 0

=> x²+3x-2x-6 = 0

=> x(x+3)-2(x+3) = 0

=> (x+3)(x-2) = 0

=> x+3 = 0 (or) x-2 = 0

=> x = -3 (or) x = 2

The roots are -3 and 2

Quadratic formula method :-

Given equation is x²+x-6 = 0

On comparing with the standard quadratic equation ax²+bx+c = 0 then

a = 1

b = 1

c = -6

We know that

by Sridharacharya formula

x = [ -b±√(b²-4ac)]/2a

x = [-1±√{1²-4(1)(-6)}]/2(1)

=> x = [-1±√(1+24)]/2

=> x = [-1±√25]/2

=> x = (-1±5)/2

=> x = (-1+5)/2 (or) (-1-5)/2

=> x = 4/2 (or) -6/2

=> x = 2 (or) -3

The roots are -3 and 2

Completing the square method :-

Given equation is x²+x-6 = 0

=> x²+x = 6

=> x²+(2/2)x = 6

=> x²+2(x)(1/2) = 6

On adding (1/2)² both sides then

=> x²+2(x)(1/2)+(1/2)² = 6+(1/2)²

=> [ x+(1/2)]² = 6+(1/4)

Since, (a+b)² = a²+2ab+b²

Where, a = x and b = 1/2

=> [x+(1/2)]² = (24+1)/4

=> [x+(1/2)]² = 25/4

=> x+(1/2) = ±√(25/4)

=> x+(1/2) = ±(5/2)

=> x = ±(5/2)-(1/2)

=> x = (±5-1)/2

=> x = (5-1)/2 (or) (-5-1)/2

=> x = 4/2 (or) -6/2

=> x = 2 (or) -3

Therefore,The roots are -3 and 2

Answer 2

Question:-

Solve the quadratic equations x² + x – 6 = 0 by any methods.

Given:-

• Quadratic equations x² + x – 6 = 0.

To Solve:-

• The given quadratic equations by any methods.

Solution:-

In a quadratic equation ax² + bx + c = 0, to solve one splits the middle term in two parts so that their sum is b and product is ac.

Hence, in x² + x − 6 = 0 one needs to split 

1 × (− 6) = − 6 in two parts whose sum is 1. It is

apparent that these are 3 and −2, Hence 

x² + x − 6 = 0 can be written as

x² + 3x – 2x – 6 = 0.

x (x + 3) – 2 (x + 3) = 0.

i.e, either x – 2 = 0, i.e, x = 2 (or)

x + 3 = 0, i.e, x = – 3.

Answer:-

The roots are x = – 3 and x = 2.

Hope you have satisfied. ⚘