x^2+10x-1=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} + 10 x - 1 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 10

and

c = -1

Δ = 10^{2} - 4 \cdot 1 \cdot (-1)

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

10^{2}

10 \cdot 10

100

Δ = 100 - 4 \cdot (-1)

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

- 4 \cdot (-1)

4

Δ = 100 + 4

ⓘ Start from

100

and move forward

4

on the number line:

100 + 4

104

Δ = 104

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 10 - \sqrt{104}}{2 \cdot 1}

x_{1} = \frac{- 10 - \sqrt{104}}{2}

ⓘ We look for the largest square dividing

104

4

2^{2}

, and take it out of the root:

\sqrt{104}

\sqrt{4 \cdot 26}

\sqrt{4} \cdot \sqrt{26}

2 \cdot \sqrt{26}

■The minus sign is factored out, changing the sign of each term:

- 10 - 2 \cdot \sqrt{26}

- (10 + 2 \cdot \sqrt{26})

x_{1} = - \frac{10 + 2 \cdot \sqrt{26}}{2}

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

- \frac{10 + 2 \cdot \sqrt{26}}{2}

- \frac{2 \cdot (5 + \sqrt{26})}{2}

- (5 + \sqrt{26})

x_{1} = - (5 + \sqrt{26})

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (5 + \sqrt{26})

- 5 - \sqrt{26}

x_{1} = - 5 - \sqrt{26}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 10 + \sqrt{104}}{2 \cdot 1}

x_{2} = \frac{- 10 + \sqrt{104}}{2}

ⓘ We look for the largest square dividing

104

4

2^{2}

, and take it out of the root:

\sqrt{104}

\sqrt{4 \cdot 26}

\sqrt{4} \cdot \sqrt{26}

2 \cdot \sqrt{26}

x_{2} = \frac{- 10 + 2 \cdot \sqrt{26}}{2}

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

\frac{- 10 + 2 \cdot \sqrt{26}}{2}

\frac{2 \cdot (- 5 + \sqrt{26})}{2}

- 5 + \sqrt{26}

x_{2} = - 5 + \sqrt{26}

▶Solutions found:

x = {- \sqrt{26} - 5, \sqrt{26} - 5}

2) The numeric value of:

x = {- \sqrt{26} - 5, \sqrt{26} - 5}

is:

x = {\approx - 5.0990 - 5, \approx 5.0990 - 5}

▷

x = {\approx -10.0990, \approx 9.902 \times 10^{- 2}}