-x^2-3x+3=0 — step-by-step solution

1) Let's solve in

ℝ

- x^{2} - 3 x + 3 = 0

Since

-1 \neq 0

, divide both sides by

-1

x^{2} + 3 x - 3 = 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 = 3

and

c = -3

Δ = 3^{2} - 4 \cdot 1 \cdot (-3)

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

3^{2}

3 \cdot 3

9

Δ = 9 - 4 \cdot (-3)

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

- 4 \cdot (-3)

12

Δ = 9 + 12

ⓘ Start from

9

and move forward

12

on the number line:

9 + 12

21

Δ = 21

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 3 - \sqrt{21}}{2 \cdot 1}

x_{1} = \frac{- 3 - \sqrt{21}}{2}

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

- 3 - \sqrt{21}

- (3 + \sqrt{21})

x_{1} = - \frac{3 + \sqrt{21}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 3 + \sqrt{21}}{2 \cdot 1}

x_{2} = \frac{- 3 + \sqrt{21}}{2}

▶Solutions obtained:

x = {- \frac{3 + \sqrt{21}}{2}, \frac{\sqrt{21} - 3}{2}}

2) The numeric value of:

x = {- \frac{3 + \sqrt{21}}{2}, \frac{\sqrt{21} - 3}{2}}

is:

x = {\approx - \frac{3 + 4.5826}{2}, \approx \frac{4.5826 - 3}{2}}

x = {\approx - \frac{7.5826}{2}, \approx \frac{1.5826}{2}}

▷

x = {\approx -3.7913, \approx 0.7913}