x^2-33x+125=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - 33 x + 125 = 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 = -33

and

c = 125

Δ = {(-33)}^{2} - 4 \cdot 1 \cdot 125

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

{(-33)}^{2}

(-33) \cdot (-33)

1089

Δ = 1089 - 4 \cdot 125

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 125

4 \cdot (100 + 20 + 5)

4 \cdot 100 + 4 \cdot 20 + 4 \cdot 5

400 + 80 + 20

500

Δ = 1089 - 500

ⓘ Start from

1089

and move back

500

on the number line:

1089 - 500

589

Δ = 589

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-33) - \sqrt{589}}{2 \cdot 1}

x_{1} = \frac{33 - \sqrt{589}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-33) + \sqrt{589}}{2 \cdot 1}

x_{2} = \frac{33 + \sqrt{589}}{2}

▶Solutions obtained:

x = {\frac{33 - \sqrt{589}}{2}, \frac{33 + \sqrt{589}}{2}}

2) The numeric value of:

x = {\frac{33 - \sqrt{589}}{2}, \frac{33 + \sqrt{589}}{2}}

is:

x = {\approx \frac{33 - 24.2693}{2}, \approx \frac{33 + 24.2693}{2}}

x = {\approx \frac{8.7307}{2}, \approx \frac{57.2693}{2}}

▷

x = {\approx 4.3653, \approx 28.6347}