2x^2-x-3<0 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} - 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 = 2

b = -1

and

c = -3

Δ = {(-1)}^{2} - 4 \cdot 2 \cdot (-3)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

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

- 8 \cdot (-3)

8 \cdot 3

Δ = 1 + 8 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 3

24

Δ = 1 + 24

ⓘ Start from

1

and move forward

24

on the number line:

1 + 24

25

Δ = 25

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{25}}{2 \cdot 2}

x_{1} = \frac{1 - \sqrt{25}}{2 \cdot 2}

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

\sqrt{25}

5

;

5^{2}

25

ⅰ

2 \cdot 2

4

x_{1} = \frac{1 - 5}{4}

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

1 - 5

-4

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

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

4

, their greatest common divisor:

- \frac{4}{4}

-1

x_{1} = -1

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{25}}{2 \cdot 2}

x_{2} = \frac{1 + \sqrt{25}}{2 \cdot 2}

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

\sqrt{25}

5

;

5^{2}

25

ⅰ

2 \cdot 2

4

x_{2} = \frac{1 + 5}{4}

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

1 + 5

6

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

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

2

, their greatest common divisor:

\frac{6}{4}

\frac{3 \cdot 2}{2 \cdot 2}

\frac{3}{2}

x_{2} = \frac{3}{2}

The quadratic has the opposite sign to a =

2

between its roots, so it is indeed negative between its roots; the roots, where it is zero, are excluded.

▶Solution obtained:

x \in (-1, \frac{3}{2})

2) The numeric value of:

x \in (-1, \frac{3}{2})

equals:

▷

x \in (-1, 1.5)