x^2-9x+5=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - 9 x + 5 = 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 = -9

and

c = 5

Δ = {(-9)}^{2} - 4 \cdot 1 \cdot 5

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

{(-9)}^{2}

(-9) \cdot (-9)

81

Δ = 81 - 4 \cdot 5

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 5

20

Δ = 81 - 20

ⓘ Start from

81

and move back

20

on the number line:

81 - 20

61

Δ = 61

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-9) - \sqrt{61}}{2 \cdot 1}

x_{1} = \frac{9 - \sqrt{61}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-9) + \sqrt{61}}{2 \cdot 1}

x_{2} = \frac{9 + \sqrt{61}}{2}

▶Solutions found:

x = {\frac{9 - \sqrt{61}}{2}, \frac{9 + \sqrt{61}}{2}}

2) The numeric value of:

x = {\frac{9 - \sqrt{61}}{2}, \frac{9 + \sqrt{61}}{2}}

is:

x = {\approx \frac{9 - 7.8102}{2}, \approx \frac{9 + 7.8102}{2}}

x = {\approx \frac{1.1898}{2}, \approx \frac{16.8102}{2}}

▷

x = {\approx 0.5949, \approx 8.4051}