x^2-x-11=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - x - 11 = 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 = -1

and

c = -11

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-11)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

Δ = 1 - 4 \cdot (-11)

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

- 4 \cdot (-11)

4 \cdot 11

4 \cdot (10 + 1)

4 \cdot 10 + 4

40 + 4

44

Δ = 1 + 44

ⓘ Start from

1

and move forward

44

on the number line:

1 + 44

45

Δ = 45

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{45}}{2 \cdot 1}

x_{1} = \frac{1 - \sqrt{45}}{2}

ⓘ We look for the largest square dividing

45

9

3^{2}

, and take it out of the root:

\sqrt{45}

\sqrt{9 \cdot 5}

\sqrt{9} \cdot \sqrt{5}

3 \cdot \sqrt{5}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{45}}{2 \cdot 1}

x_{2} = \frac{1 + \sqrt{45}}{2}

ⓘ We look for the largest square dividing

45

9

3^{2}

, and take it out of the root:

\sqrt{45}

\sqrt{9 \cdot 5}

\sqrt{9} \cdot \sqrt{5}

3 \cdot \sqrt{5}

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

▶Solutions found:

x = {\frac{1 - 3 \cdot \sqrt{5}}{2}, \frac{1 + 3 \cdot \sqrt{5}}{2}}

2) The numeric value of:

x = {\frac{1 - 3 \cdot \sqrt{5}}{2}, \frac{1 + 3 \cdot \sqrt{5}}{2}}

equals:

x = {\approx \frac{1 - 3 \cdot 2.2361}{2}, \approx \frac{1 + 3 \cdot 2.2361}{2}}

x = {\approx \frac{1 - 6.7082}{2}, \approx \frac{1 + 6.7082}{2}}

x = {\approx - \frac{5.7082}{2}, \approx \frac{7.7082}{2}}

▷

x = {\approx -2.8541, \approx 3.8541}