x^2-9x-5x-2=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2}

- 9 x

- 5 x

- 2 = 0

ⓘ Like terms are grouped by adding their coefficients:

- 9 x - 5 x

-14 x

x^{2} - 14 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 = 1

b = -14

and

c = -2

Δ = {(-14)}^{2} - 4 \cdot 1 \cdot (-2)

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

{(-14)}^{2}

(-14) \cdot (-14)

196

Δ = 196 - 4 \cdot (-2)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-2)

8

Δ = 196 + 8

ⓘ Start from

196

and move forward

8

on the number line:

196 + 8

204

Δ = 204

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-14) - \sqrt{204}}{2 \cdot 1}

x_{1} = \frac{14 - \sqrt{204}}{2}

ⓘ We look for the largest square dividing

204

4

2^{2}

, and take it out of the root:

\sqrt{204}

\sqrt{4 \cdot 51}

\sqrt{4} \cdot \sqrt{51}

2 \cdot \sqrt{51}

x_{1} = \frac{14 - 2 \cdot \sqrt{51}}{2}

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

\frac{14 - 2 \cdot \sqrt{51}}{2}

\frac{2 \cdot (7 - \sqrt{51})}{2}

7 - \sqrt{51}

x_{1} = 7 - \sqrt{51}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-14) + \sqrt{204}}{2 \cdot 1}

x_{2} = \frac{14 + \sqrt{204}}{2}

ⓘ We look for the largest square dividing

204

4

2^{2}

, and take it out of the root:

\sqrt{204}

\sqrt{4 \cdot 51}

\sqrt{4} \cdot \sqrt{51}

2 \cdot \sqrt{51}

x_{2} = \frac{14 + 2 \cdot \sqrt{51}}{2}

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

\frac{14 + 2 \cdot \sqrt{51}}{2}

\frac{2 \cdot (7 + \sqrt{51})}{2}

7 + \sqrt{51}

x_{2} = 7 + \sqrt{51}

▶Solutions found:

x = {7 - \sqrt{51}, 7 + \sqrt{51}}

2) The numeric value of:

x = {7 - \sqrt{51}, 7 + \sqrt{51}}

is:

x = {\approx 7 - 7.1414, \approx 7 + 7.1414}

▷

x = {\approx -0.1414, \approx 14.1414}