6x^2-4x-5=4x — step-by-step solution

1) Let's solve in

ℝ

6 x^{2} - 4 x - 5 = 4 x

6 x^{2}

- 4 x

- 5

- 4 x = 0

ⓘ Like terms are grouped by adding their coefficients:

- 4 x - 4 x

-8 x

6 x^{2} - 8 x - 5 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 6

b = -8

and

c = -5

Δ = {(-8)}^{2} - 4 \cdot 6 \cdot (-5)

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

{(-8)}^{2}

(-8) \cdot (-8)

64

■Multiplications are done before additions and subtractions:

4 \cdot 6

24

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

- 24 \cdot (-5)

24 \cdot 5

Δ = 64 + 24 \cdot 5

ⓘ Multiplications are done before additions and subtractions:

24 \cdot 5

(20 + 4) \cdot 5

20 \cdot 5 + 4 \cdot 5

100 + 20

120

Δ = 64 + 120

ⓘ Start from

64

and move forward

120

on the number line:

64 + 120

184

Δ = 184

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-8) - \sqrt{184}}{2 \cdot 6}

x_{1} = \frac{8 - \sqrt{184}}{2 \cdot 6}

ⓘ We look for the largest square dividing

184

4

2^{2}

, and take it out of the root:

\sqrt{184}

\sqrt{4 \cdot 46}

\sqrt{4} \cdot \sqrt{46}

2 \cdot \sqrt{46}

ⅰ

2 \cdot 6

12

x_{1} = \frac{8 - 2 \cdot \sqrt{46}}{12}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{8 - 2 \cdot \sqrt{46}}{12}

\frac{4 - \sqrt{46}}{6}

x_{1} = \frac{4 - \sqrt{46}}{6}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-8) + \sqrt{184}}{2 \cdot 6}

x_{2} = \frac{8 + \sqrt{184}}{2 \cdot 6}

ⓘ We look for the largest square dividing

184

4

2^{2}

, and take it out of the root:

\sqrt{184}

\sqrt{4 \cdot 46}

\sqrt{4} \cdot \sqrt{46}

2 \cdot \sqrt{46}

ⅰ

2 \cdot 6

12

x_{2} = \frac{8 + 2 \cdot \sqrt{46}}{12}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{8 + 2 \cdot \sqrt{46}}{12}

\frac{4 + \sqrt{46}}{6}

x_{2} = \frac{4 + \sqrt{46}}{6}

▶Solutions obtained:

x = {\frac{4 - \sqrt{46}}{6}, \frac{4 + \sqrt{46}}{6}}

2) The numeric value of:

x = {\frac{4 - \sqrt{46}}{6}, \frac{4 + \sqrt{46}}{6}}

is:

x = {\approx \frac{4 - 6.7823}{6}, \approx \frac{4 + 6.7823}{6}}

x = {\approx - \frac{2.7823}{6}, \approx \frac{10.7823}{6}}

▷

x = {\approx -0.4637, \approx 1.7971}