x^2+16x+62=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} + 16 x + 62 = 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 = 16

and

c = 62

Δ = 16^{2} - 4 \cdot 1 \cdot 62

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

16^{2}

16 \cdot 16

256

Δ = 256 - 4 \cdot 62

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 62

4 \cdot (60 + 2)

4 \cdot 60 + 4 \cdot 2

240 + 8

248

Δ = 256 - 248

ⓘ Start from

256

and move back

248

on the number line:

256 - 248

8

Δ = 8

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 16 - \sqrt{8}}{2 \cdot 1}

x_{1} = \frac{- 16 - \sqrt{8}}{2}

ⓘ We look for the largest square dividing

8

4

2^{2}

, and take it out of the root:

\sqrt{8}

\sqrt{4 \cdot 2}

\sqrt{4} \cdot \sqrt{2}

2 \cdot \sqrt{2}

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

- 16 - 2 \cdot \sqrt{2}

- (16 + 2 \cdot \sqrt{2})

x_{1} = - \frac{16 + 2 \cdot \sqrt{2}}{2}

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

- \frac{16 + 2 \cdot \sqrt{2}}{2}

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

- (8 + \sqrt{2})

x_{1} = - (8 + \sqrt{2})

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

- (8 + \sqrt{2})

- 8 - \sqrt{2}

x_{1} = - 8 - \sqrt{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 16 + \sqrt{8}}{2 \cdot 1}

x_{2} = \frac{- 16 + \sqrt{8}}{2}

ⓘ We look for the largest square dividing

8

4

2^{2}

, and take it out of the root:

\sqrt{8}

\sqrt{4 \cdot 2}

\sqrt{4} \cdot \sqrt{2}

2 \cdot \sqrt{2}

x_{2} = \frac{- 16 + 2 \cdot \sqrt{2}}{2}

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

\frac{- 16 + 2 \cdot \sqrt{2}}{2}

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

- 8 + \sqrt{2}

x_{2} = - 8 + \sqrt{2}

▶Solutions obtained:

x = {- \sqrt{2} - 8, \sqrt{2} - 8}

2) The numeric value of:

x = {- \sqrt{2} - 8, \sqrt{2} - 8}

equals:

x = {\approx - 1.4142 - 8, \approx 1.4142 - 8}

▷

x = {\approx -9.4142, \approx -6.5858}