-x^2+6x+3=0 — step-by-step solution

1) Let's solve in

ℝ

- x^{2} + 6 x + 3 = 0

Since

-1 \neq 0

, divide both sides by

-1

x^{2} - 6 x - 3 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -6

and

c = -3

Δ = {(-6)}^{2} - 4 \cdot 1 \cdot (-3)

ⓘ 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 (-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 = 36 + 12

ⓘ Start from

36

and move forward

12

on the number line:

36 + 12

48

Δ = 48

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-6) - \sqrt{48}}{2 \cdot 1}

x_{1} = \frac{6 - \sqrt{48}}{2}

ⓘ We look for the largest square dividing

48

16

4^{2}

, and take it out of the root:

\sqrt{48}

\sqrt{16 \cdot 3}

\sqrt{16} \cdot \sqrt{3}

4 \cdot \sqrt{3}

x_{1} = \frac{6 - 4 \cdot \sqrt{3}}{2}

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

\frac{6 - 4 \cdot \sqrt{3}}{2}

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

3 - 2 \cdot \sqrt{3}

x_{1} = 3 - 2 \cdot \sqrt{3}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-6) + \sqrt{48}}{2 \cdot 1}

x_{2} = \frac{6 + \sqrt{48}}{2}

ⓘ We look for the largest square dividing

48

16

4^{2}

, and take it out of the root:

\sqrt{48}

\sqrt{16 \cdot 3}

\sqrt{16} \cdot \sqrt{3}

4 \cdot \sqrt{3}

x_{2} = \frac{6 + 4 \cdot \sqrt{3}}{2}

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

\frac{6 + 4 \cdot \sqrt{3}}{2}

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

3 + 2 \cdot \sqrt{3}

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

▶Solutions found:

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

2) The numeric value of:

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

is:

x = {\approx 3 - 2 \cdot 1.7321, \approx 3 + 2 \cdot 1.7321}

x = {\approx 3 - 3.4641, \approx 3 + 3.4641}

▷

x = {\approx -0.4641, \approx 6.4641}