x^2-8x-105=-10 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - 8 x - 105 = -10

ⓘ We add

10

to both sides

x^{2}

- 8 x

- 105

+ 10 = 0

ⓘ Start from

-105

and move forward

10

on the number line, getting closer to zero:

- 105 + 10

-95

x^{2} - 8 x - 95 = 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 = -8

and

c = -95

Δ = {(-8)}^{2} - 4 \cdot 1 \cdot (-95)

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

{(-8)}^{2}

(-8) \cdot (-8)

64

Δ = 64 - 4 \cdot (-95)

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

- 4 \cdot (-95)

4 \cdot 95

4 \cdot (90 + 5)

4 \cdot 90 + 4 \cdot 5

360 + 20

380

Δ = 64 + 380

ⓘ Start from

64

and move forward

380

on the number line:

64 + 380

444

Δ = 444

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-8) - \sqrt{444}}{2 \cdot 1}

x_{1} = \frac{8 - \sqrt{444}}{2}

ⓘ We look for the largest square dividing

444

4

2^{2}

, and take it out of the root:

\sqrt{444}

\sqrt{4 \cdot 111}

\sqrt{4} \cdot \sqrt{111}

2 \cdot \sqrt{111}

x_{1} = \frac{8 - 2 \cdot \sqrt{111}}{2}

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

\frac{8 - 2 \cdot \sqrt{111}}{2}

\frac{2 \cdot (4 - \sqrt{111})}{2}

4 - \sqrt{111}

x_{1} = 4 - \sqrt{111}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-8) + \sqrt{444}}{2 \cdot 1}

x_{2} = \frac{8 + \sqrt{444}}{2}

ⓘ We look for the largest square dividing

444

4

2^{2}

, and take it out of the root:

\sqrt{444}

\sqrt{4 \cdot 111}

\sqrt{4} \cdot \sqrt{111}

2 \cdot \sqrt{111}

x_{2} = \frac{8 + 2 \cdot \sqrt{111}}{2}

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

\frac{8 + 2 \cdot \sqrt{111}}{2}

\frac{2 \cdot (4 + \sqrt{111})}{2}

4 + \sqrt{111}

x_{2} = 4 + \sqrt{111}

▶Solutions found:

x = {4 - \sqrt{111}, 4 + \sqrt{111}}

2) The numeric value of:

x = {4 - \sqrt{111}, 4 + \sqrt{111}}

equals:

x = {\approx 4 - 10.5357, \approx 4 + 10.5357}

▷

x = {\approx -6.5357, \approx 14.5357}