x^2-9x+9=0 — step-by-step solution

1) Let's solve in

ℝ

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

and

c = 9

Δ = {(-9)}^{2} - 4 \cdot 1 \cdot 9

ⓘ 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 9

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 9

36

Δ = 81 - 36

ⓘ Start from

81

and move back

36

on the number line:

81 - 36

45

Δ = 45

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-9) - \sqrt{45}}{2 \cdot 1}

x_{1} = \frac{9 - \sqrt{45}}{2}

ⓘ We look for the largest square dividing

45

9

3^{2}

, and take it out of the root:

\sqrt{45}

\sqrt{9 \cdot 5}

\sqrt{9} \cdot \sqrt{5}

3 \cdot \sqrt{5}

x_{1} = \frac{9 - 3 \cdot \sqrt{5}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-9) + \sqrt{45}}{2 \cdot 1}

x_{2} = \frac{9 + \sqrt{45}}{2}

ⓘ We look for the largest square dividing

45

9

3^{2}

, and take it out of the root:

\sqrt{45}

\sqrt{9 \cdot 5}

\sqrt{9} \cdot \sqrt{5}

3 \cdot \sqrt{5}

x_{2} = \frac{9 + 3 \cdot \sqrt{5}}{2}

▶Solutions obtained:

x = {\frac{9 - 3 \cdot \sqrt{5}}{2}, \frac{9 + 3 \cdot \sqrt{5}}{2}}

2) The numeric value of:

x = {\frac{9 - 3 \cdot \sqrt{5}}{2}, \frac{9 + 3 \cdot \sqrt{5}}{2}}

equals:

x = {\approx \frac{9 - 3 \cdot 2.2361}{2}, \approx \frac{9 + 3 \cdot 2.2361}{2}}

x = {\approx \frac{9 - 6.7082}{2}, \approx \frac{9 + 6.7082}{2}}

x = {\approx \frac{2.2918}{2}, \approx \frac{15.7082}{2}}

▷

x = {\approx 1.1459, \approx 7.8541}