2n^2+5n+3=0 — step-by-step solution

1) Let's solve in

ℝ

2 n^{2} + 5 n + 3 = 0

Given the quadratic form

a n^{2} + b n + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 5

and

c = 3

Δ = 5^{2} - 4 \cdot 2 \cdot 3

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

5^{2}

5 \cdot 5

25

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 25 - 8 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 3

24

Δ = 25 - 24

ⓘ Start from

25

and move back

24

on the number line:

25 - 24

1

Δ = 1

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

n_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 5 - \sqrt{1}}{2 \cdot 2}

n_{1} = \frac{- 5 - 1}{2 \cdot 2}

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

- 5 - 1

-6

ⅰ

2 \cdot 2

4

n_{1} = - \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}

n_{1} = - \frac{3}{2}

The quadratic formula gives

n_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 5 + \sqrt{1}}{2 \cdot 2}

n_{2} = \frac{- 5 + 1}{2 \cdot 2}

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

- 5 + 1

-4

ⅰ

2 \cdot 2

4

n_{2} = - \frac{4}{4}

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

4

, their greatest common divisor:

- \frac{4}{4}

-1

n_{2} = -1

▶Solutions found:

n = {- \frac{3}{2}, -1}

2) The numeric value of:

n = {- \frac{3}{2}, -1}

is:

▷

n = {-1.5, -1}