x^2+5x+1=-5 — step-by-step solution

Let's solve in

ℝ

x^{2} + 5 x + 1 = -5

ⓘ We add

5

to both sides

x^{2}

+ 5 x

+ 1

+ 5 = 0

ⓘ Start from

1

and move forward

5

on the number line:

1 + 5

6

x^{2} + 5 x + 6 = 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 = 5

and

c = 6

Δ = 5^{2} - 4 \cdot 1 \cdot 6

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

5^{2}

5 \cdot 5

25

Δ = 25 - 4 \cdot 6

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 6

24

Δ = 25 - 24

ⓘ Start from

25

and move back

24

on the number line:

25 - 24

1

Δ = 1

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 5 - \sqrt{1}}{2 \cdot 1}

x_{1} = \frac{- 5 - 1}{2}

ⓘ The numerator and the denominator are computed before the division:

- 5 - 1

-6

x_{1} = - \frac{6}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{6}{2}

-3

x_{1} = -3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 5 + \sqrt{1}}{2 \cdot 1}

x_{2} = \frac{- 5 + 1}{2}

ⓘ The numerator and the denominator are computed before the division:

- 5 + 1

-4

x_{2} = - \frac{4}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{4}{2}

-2

x_{2} = -2

▶Solutions obtained:

x = {-3, -2}