(x+1)^2=2(x-1)^2 — step-by-step solution | Explain MAths

(x+1)^2=2(x-1)^2

1) Let's solve in

ℝ

:

{(x + 1)}^{2} = 2 \cdot {(x - 1)}^{2}

{(x + 1)}^{2} - 2 \cdot {(x - 1)}^{2} = 0

ℹ

{(x + 1)}^{2}

=

(x + 1) \cdot (x + 1)

‖

{(x - 1)}^{2}

=

(x - 1) \cdot (x - 1)

(x + 1) \cdot (x + 1) - 2 \cdot (x - 1) \cdot (x - 1) = 0

ℹ

(x + 1) \cdot (x + 1)

=

x \cdot x

+ x \cdot 1

+ 1 \cdot x

+ 1 \cdot 1

‖

2 \cdot (x - 1)

=

2 x - 2

(x \cdot x

+ x \cdot 1

+ 1 \cdot x

+ 1 \cdot 1) - (2 x - 2) \cdot (x - 1) = 0

ℹ

x \cdot x

=

x^{2}

‖

1 \cdot 1

=

1

x^{2}

+ x

+ x

+ 1

- (2 x - 2) \cdot (x - 1) = 0

x^{2}

+ 2 x

+ 1

- (2 x - 2) \cdot (x - 1) = 0

x^{2}

+ 2 x

+ 1

- (2 \cdot x \cdot x

+ 2 x \cdot (-1)

- 2 x

- 2 \cdot (-1)) = 0

x^{2}

+ 2 x

+ 1

- 2 \cdot x \cdot x

- 2 x \cdot (-1)

+ 2 x

+ 2 \cdot (-1) = 0

ℹ

x \cdot x

=

x^{2}

‖

-2 \cdot (-1)

=

2

‖

2 x + 2 x

=

4 x

‖

2 \cdot (-1)

=

-2

x^{2}

+ 4 x

+ 1

- 2 x^{2}

+ 2 x

- 2 = 0

ℹ

1 - 2

=

-1

‖

x^{2} - 2 x^{2}

=

- x^{2}

‖

4 x + 2 x

=

6 x

- x^{2} + 6 x - 1 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

Δ = 6^{2} - 4 \cdot (-1) \cdot (-1)

ℹ

6^{2}

=

36

‖

-4 \cdot (-1)

=

4

‖

4 \cdot (-1)

=

-4

Δ = 36 - 4

Δ = 32

Since

Δ > 0

, the equation has two solutions:

Solution 1st:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{- 6 - \sqrt{32}}{2 \cdot (-1)}

ℹ

- \sqrt{32}

=

- 4 \cdot \sqrt{2}

‖

2 \cdot (-1)

=

-2

x1 = \frac{- (6 + 4 \cdot \sqrt{2})}{-2}

x1 = \frac{6 + 4 \cdot \sqrt{2}}{2}

x1 = 3 + 2 \cdot \sqrt{2}

For the 2nd solution, we obtain:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{- 6 + \sqrt{32}}{2 \cdot (-1)}

ℹ

\sqrt{32}

=

4 \cdot \sqrt{2}

‖

2 \cdot (-1)

=

-2

x2 = \frac{- 6 + 4 \cdot \sqrt{2}}{-2}

x2 = - \frac{- 6 + 4 \cdot \sqrt{2}}{2}

x2 = \frac{6 - 4 \cdot \sqrt{2}}{2}

x2 = 3 - 2 \cdot \sqrt{2}

Solution obtained:

x = 3 - 2 \cdot \sqrt{2}

Solution obtained:

x = 2 \cdot \sqrt{2} + 3

▶Solution:

x = {3 - 2 \cdot \sqrt{2}, 2 \cdot \sqrt{2} + 3}

2) The numeric value of:

x = {3 - 2 \cdot \sqrt{2}, 2 \cdot \sqrt{2} + 3}

is:

x = {3 - 2 \cdot \approx 1.4142, 2 \cdot \approx 1.4142 + 3}

x = {3 + \approx -2.8284, \approx 2.8284 + 3}

▷

x = {\approx 0.1716, \approx 5.8284}