3x^2-5x=-2 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} - 5 x = -2

3 x^{2} - 5 x + 2 = 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 = -5

and

c = 2

Δ = {(-5)}^{2} - 4 \cdot 3 \cdot 2

ⓘ 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 3

12

Δ = 25 - 12 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

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

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-5) - \sqrt{1}}{2 \cdot 3}

x_{1} = \frac{5 - 1}{2 \cdot 3}

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

5 - 1

4

ⅰ

2 \cdot 3

6

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

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

2

, their greatest common divisor:

\frac{4}{6}

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

\frac{2}{3}

x_{1} = \frac{2}{3}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-5) + \sqrt{1}}{2 \cdot 3}

x_{2} = \frac{5 + 1}{2 \cdot 3}

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

5 + 1

6

ⅰ

2 \cdot 3

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 found:

x = {\frac{2}{3}, 1}

2) The numeric value of:

x = {\frac{2}{3}, 1}

equals:

▷

x = {\approx 0.6667, 1}