3x^2+x-8=0 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} + x - 8 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 3

b = 1

and

c = -8

Δ = 1 - 4 \cdot 3 \cdot (-8)

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

■Two minus signs cancel out, the product of two negative numbers is positive:

- 12 \cdot (-8)

12 \cdot 8

Δ = 1 + 12 \cdot 8

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 8

(10 + 2) \cdot 8

10 \cdot 8 + 2 \cdot 8

80 + 16

96

Δ = 1 + 96

ⓘ Start from

1

and move forward

96

on the number line:

1 + 96

97

Δ = 97

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{97}}{2 \cdot 3}

x_{1} = \frac{- 1 - \sqrt{97}}{2 \cdot 3}

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

- 1 - \sqrt{97}

- (1 + \sqrt{97})

ⅰ

2 \cdot 3

6

x_{1} = - \frac{1 + \sqrt{97}}{6}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{97}}{2 \cdot 3}

x_{2} = \frac{- 1 + \sqrt{97}}{2 \cdot 3}

ⅰ

2 \cdot 3

6

x_{2} = \frac{- 1 + \sqrt{97}}{6}

▶Solutions found:

x = {- \frac{1 + \sqrt{97}}{6}, \frac{\sqrt{97} - 1}{6}}

2) The numeric value of:

x = {- \frac{1 + \sqrt{97}}{6}, \frac{\sqrt{97} - 1}{6}}

equals:

x = {\approx - \frac{1 + 9.8489}{6}, \approx \frac{9.8489 - 1}{6}}

x = {\approx - \frac{10.8489}{6}, \approx \frac{8.8489}{6}}

▷

x = {\approx -1.8081, \approx 1.4748}