x^2+4x-16=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} + 4 x - 16 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 4

and

c = -16

Δ = 4^{2} - 4 \cdot 1 \cdot (-16)

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{2}

4 \cdot 4

16

Δ = 16 - 4 \cdot (-16)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-16)

4 \cdot 16

4 \cdot (10 + 6)

4 \cdot 10 + 4 \cdot 6

40 + 24

64

Δ = 16 + 64

ⓘ Start from

16

and move forward

64

on the number line:

16 + 64

80

Δ = 80

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 4 - \sqrt{80}}{2 \cdot 1}

x_{1} = \frac{- 4 - \sqrt{80}}{2}

ⓘ We look for the largest square dividing

80

16

4^{2}

, and take it out of the root:

\sqrt{80}

\sqrt{16 \cdot 5}

\sqrt{16} \cdot \sqrt{5}

4 \cdot \sqrt{5}

■The minus sign is factored out, changing the sign of each term:

- 4 - 4 \cdot \sqrt{5}

- (4 + 4 \cdot \sqrt{5})

x_{1} = - \frac{4 + 4 \cdot \sqrt{5}}{2}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{4 + 4 \cdot \sqrt{5}}{2}

- \frac{2 \cdot (2 + 2 \cdot \sqrt{5})}{2}

- (2 + 2 \cdot \sqrt{5})

x_{1} = - (2 + 2 \cdot \sqrt{5})

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (2 + 2 \cdot \sqrt{5})

- 2 - 2 \cdot \sqrt{5}

x_{1} = - 2 - 2 \cdot \sqrt{5}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 4 + \sqrt{80}}{2 \cdot 1}

x_{2} = \frac{- 4 + \sqrt{80}}{2}

ⓘ We look for the largest square dividing

80

16

4^{2}

, and take it out of the root:

\sqrt{80}

\sqrt{16 \cdot 5}

\sqrt{16} \cdot \sqrt{5}

4 \cdot \sqrt{5}

x_{2} = \frac{- 4 + 4 \cdot \sqrt{5}}{2}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{- 4 + 4 \cdot \sqrt{5}}{2}

\frac{2 \cdot (- 2 + 2 \cdot \sqrt{5})}{2}

- 2 + 2 \cdot \sqrt{5}

x_{2} = - 2 + 2 \cdot \sqrt{5}

▶Solutions obtained:

x = {- 2 \cdot \sqrt{5} - 2, 2 \cdot \sqrt{5} - 2}

2) The numeric value of:

x = {- 2 \cdot \sqrt{5} - 2, 2 \cdot \sqrt{5} - 2}

is:

x = {\approx - 2 \cdot 2.2361 - 2, \approx 2 \cdot 2.2361 - 2}

x = {\approx - 4.4721 - 2, \approx 4.4721 - 2}

▷

x = {\approx -6.4721, \approx 2.4721}