30x^2+21x+3=0 — step-by-step solution

1) Let's solve in

ℝ

30 x^{2} + 21 x + 3 = 0

Since

3 \neq 0

, divide both sides by

3

10 x^{2} + 7 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 = 10

b = 7

and

c = 1

Δ = 7^{2} - 4 \cdot 10 \cdot 1

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

7^{2}

7 \cdot 7

49

Δ = 49 - 4 \cdot 10

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 10

40

Δ = 49 - 40

ⓘ Start from

49

and move back

40

on the number line:

49 - 40

9

Δ = 9

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 7 - \sqrt{9}}{2 \cdot 10}

x_{1} = \frac{- 7 - \sqrt{9}}{2 \cdot 10}

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 10

20

x_{1} = \frac{- 7 - 3}{20}

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

- 7 - 3

-10

x_{1} = - \frac{10}{20}

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

10

, their greatest common divisor:

- \frac{10}{20}

- \frac{1 \cdot 10}{2 \cdot 10}

- \frac{1}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 7 + \sqrt{9}}{2 \cdot 10}

x_{2} = \frac{- 7 + \sqrt{9}}{2 \cdot 10}

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 10

20

x_{2} = \frac{- 7 + 3}{20}

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

- 7 + 3

-4

x_{2} = - \frac{4}{20}

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

4

, their greatest common divisor:

- \frac{4}{20}

- \frac{1 \cdot 4}{5 \cdot 4}

- \frac{1}{5}

x_{2} = - \frac{1}{5}

▶Solutions found:

x = {- \frac{1}{2}, - \frac{1}{5}}

2) The numeric value of:

x = {- \frac{1}{2}, - \frac{1}{5}}

equals:

▷

x = {-0.5, -0.2}