3x^2-7x-5=0 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} - 7 x - 5 = 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 = -7

and

c = -5

Δ = {(-7)}^{2} - 4 \cdot 3 \cdot (-5)

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

{(-7)}^{2}

(-7) \cdot (-7)

49

■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 (-5)

12 \cdot 5

Δ = 49 + 12 \cdot 5

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 5

(10 + 2) \cdot 5

10 \cdot 5 + 2 \cdot 5

50 + 10

60

Δ = 49 + 60

ⓘ Start from

49

and move forward

60

on the number line:

49 + 60

109

Δ = 109

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-7) - \sqrt{109}}{2 \cdot 3}

x_{1} = \frac{7 - \sqrt{109}}{2 \cdot 3}

ⅰ

2 \cdot 3

6

x_{1} = \frac{7 - \sqrt{109}}{6}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-7) + \sqrt{109}}{2 \cdot 3}

x_{2} = \frac{7 + \sqrt{109}}{2 \cdot 3}

ⅰ

2 \cdot 3

6

x_{2} = \frac{7 + \sqrt{109}}{6}

▶Solutions obtained:

x = {\frac{7 - \sqrt{109}}{6}, \frac{7 + \sqrt{109}}{6}}

2) The numeric value of:

x = {\frac{7 - \sqrt{109}}{6}, \frac{7 + \sqrt{109}}{6}}

is:

x = {\approx \frac{7 - 10.4403}{6}, \approx \frac{7 + 10.4403}{6}}

x = {\approx - \frac{3.4403}{6}, \approx \frac{17.4403}{6}}

▷

x = {\approx -0.5734, \approx 2.9067}