3x^2-2x-1=0 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} - 2 x - 1 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 3

b = -2

and

c = -1

Δ = {(-2)}^{2} - 4 \cdot 3 \cdot (-1)

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

{(-2)}^{2}

(-2) \cdot (-2)

4

■Multiplications are done before additions and subtractions:

4 \cdot 3

12

■Multiplying by -1 gives the opposite:

- 12 \cdot (-1)

12

Δ = 4 + 12

ⓘ Start from

4

and move forward

12

on the number line:

4 + 12

16

Δ = 16

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-2) - \sqrt{16}}{2 \cdot 3}

x_{1} = \frac{2 - \sqrt{16}}{2 \cdot 3}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 3

6

x_{1} = \frac{2 - 4}{6}

ⓘ The numerator and the denominator are computed before the division:

2 - 4

-2

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

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{2}{6}

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

- \frac{1}{3}

x_{1} = - \frac{1}{3}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-2) + \sqrt{16}}{2 \cdot 3}

x_{2} = \frac{2 + \sqrt{16}}{2 \cdot 3}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 3

6

x_{2} = \frac{2 + 4}{6}

ⓘ The numerator and the denominator are computed before the division:

2 + 4

6

x_{2} = \frac{6}{6}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

6

, their greatest common divisor:

\frac{6}{6}

1

x_{2} = 1

▶Solutions obtained:

x = {- \frac{1}{3}, 1}

2) The numeric value of:

x = {- \frac{1}{3}, 1}

equals:

▷

x = {\approx -0.3333, 1}