Question

. Solve X2 + 2x +1 = 0 (Qudratic equation)
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Answer 1

Answer :-

x^2 + 2x + 1

By using factorization,

x^2 + x + x + 1

x( x + 1 ) + 1( x + 1 )

Ans :( x + 1 ) ( x + 1 )

x = -1 , x = -1

α = -1 , β = -1

Quadratic formula can be represented in the form of ax2 + bx + c . Here, a = 1 , b = 2, c = 1

Verification :-

α + β = -b/a

Put the required values,

-1 + -1 = -2/1

-2 = -2

LHS = RHS

αβ = c/a

Put the required values,

-1 * -1 = 1 /1

1 = 1

LHS = RHS

Answer 2

Step-by-step explanation:

The equation

x² + 2x - 1 = 0 ………………………………………………………………………(1)

being a quadratic in x, it will have two roots. We will find them as follows.

Method 1: (By Factorization)

Rewriting (1), x² + 2x = 1

Adding 1² to the left-side and an equal amount to the right-side, we get

x² + 2x + 1² = 1 + 1²

Or, x² + 2.x.1 + 1² = 1 + 1

Or, (x + 1)² = 2

Or, x+1 = ±√2

This gives, x = √2 -1, -√2 -1

Method 2: (By Formula)

Comparing (1) with the standard quadratic equation

ax² + bx + c = 0,

we get a = 1, b = 2, c = -1

Now the formula that gives the roots or solution of a quadratic equation is

x = [-b ± √(b² - 4ac)]/2a

Substituting for a, b, c into the above formula

x = -2 ± √[2² - 4.1.(-1)]/2.1 = -2 ± √(4 + 4)/2 = (-2 ± √8)/2 =

= (-2 ± 2√2)/2 = -1 ± √2

that is x₁ = √2 - 1, x₂ == -√2 -1

Hence the two roots of the equation are √2 -1 and -√2 -1 .

Result check:

x = √2 -1 . Substituting

x² + 2x - 1 = (√2 -1)² +2(√2 -1) - 1 = (√2)² -2(√2) + 1 +2√2 -2 -1

= 2 - 2√2 + 1 +2√2 -2 -1 = 0=R.H.S.

x = -√2 -1 . Substituting

x² + 2x - 1 = (-√2 -1)² +2(-√2 -1) - 1 = (√2 +1)² -2(√2 +1) - 1

=(√2)² +2(√2) + 1² -2√2 -2 -1 = 2 +2√2 + 1 -2√2 -3 = 3–3 = 0=R.H.S.

Hence the values of roots of equation(1) are correct