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. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-9)

36

Δ = 81 + 36

ⓘ Start from

81

and move forward

36

on the number line:

81 + 36

117

Δ = 117

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 9 - \sqrt{117}}{2 \cdot 1}

x_{1} = \frac{- 9 - \sqrt{117}}{2}

ⓘ We look for the largest square dividing

117

9

3^{2}

, and take it out of the root:

\sqrt{117}

\sqrt{9 \cdot 13}

\sqrt{9} \cdot \sqrt{13}

3 \cdot \sqrt{13}

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

- 9 - 3 \cdot \sqrt{13}

- (9 + 3 \cdot \sqrt{13})

x_{1} = - \frac{9 + 3 \cdot \sqrt{13}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 9 + \sqrt{117}}{2 \cdot 1}

x_{2} = \frac{- 9 + \sqrt{117}}{2}

ⓘ We look for the largest square dividing

117

9

3^{2}

, and take it out of the root:

\sqrt{117}

\sqrt{9 \cdot 13}

\sqrt{9} \cdot \sqrt{13}

3 \cdot \sqrt{13}

x_{2} = \frac{- 9 + 3 \cdot \sqrt{13}}{2}

▶Solutions obtained:

x = {- \frac{9 + 3 \cdot \sqrt{13}}{2}, \frac{3 \cdot \sqrt{13} - 9}{2}}

2) The numeric value of:

x = {- \frac{9 + 3 \cdot \sqrt{13}}{2}, \frac{3 \cdot \sqrt{13} - 9}{2}}

is:

x = {\approx - \frac{9 + 3 \cdot 3.6056}{2}, \approx \frac{3 \cdot 3.6056 - 9}{2}}

x = {\approx - \frac{9 + 10.8167}{2}, \approx \frac{10.8167 - 9}{2}}

x = {\approx - \frac{19.8167}{2}, \approx \frac{1.8167}{2}}

▷

x = {\approx -9.9083, \approx 0.9083}