-x^2-6x+6=0 — step-by-step solution

1) Let's solve in

ℝ

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

Since

-1 \neq 0

, divide both sides by

-1

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

and

c = -6

Δ = 6^{2} - 4 \cdot 1 \cdot (-6)

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

6^{2}

6 \cdot 6

36

Δ = 36 - 4 \cdot (-6)

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

- 4 \cdot (-6)

24

Δ = 36 + 24

ⓘ Start from

36

and move forward

24

on the number line:

36 + 24

60

Δ = 60

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 6 - \sqrt{60}}{2 \cdot 1}

x_{1} = \frac{- 6 - \sqrt{60}}{2}

ⓘ We look for the largest square dividing

60

4

2^{2}

, and take it out of the root:

\sqrt{60}

\sqrt{4 \cdot 15}

\sqrt{4} \cdot \sqrt{15}

2 \cdot \sqrt{15}

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

- 6 - 2 \cdot \sqrt{15}

- (6 + 2 \cdot \sqrt{15})

x_{1} = - \frac{6 + 2 \cdot \sqrt{15}}{2}

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

- \frac{6 + 2 \cdot \sqrt{15}}{2}

- \frac{2 \cdot (3 + \sqrt{15})}{2}

- (3 + \sqrt{15})

x_{1} = - (3 + \sqrt{15})

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

- (3 + \sqrt{15})

- 3 - \sqrt{15}

x_{1} = - 3 - \sqrt{15}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 6 + \sqrt{60}}{2 \cdot 1}

x_{2} = \frac{- 6 + \sqrt{60}}{2}

ⓘ We look for the largest square dividing

60

4

2^{2}

, and take it out of the root:

\sqrt{60}

\sqrt{4 \cdot 15}

\sqrt{4} \cdot \sqrt{15}

2 \cdot \sqrt{15}

x_{2} = \frac{- 6 + 2 \cdot \sqrt{15}}{2}

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

\frac{- 6 + 2 \cdot \sqrt{15}}{2}

\frac{2 \cdot (- 3 + \sqrt{15})}{2}

- 3 + \sqrt{15}

x_{2} = - 3 + \sqrt{15}

▶Solutions obtained:

x = {- \sqrt{15} - 3, \sqrt{15} - 3}

2) The numeric value of:

x = {- \sqrt{15} - 3, \sqrt{15} - 3}

equals:

x = {\approx - 3.8730 - 3, \approx 3.8730 - 3}

▷

x = {\approx -6.8730, \approx 0.8730}