x^2+7x-27=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} + 7 x - 27 = 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 = 7

and

c = -27

Δ = 7^{2} - 4 \cdot 1 \cdot (-27)

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

7^{2}

7 \cdot 7

49

Δ = 49 - 4 \cdot (-27)

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

- 4 \cdot (-27)

4 \cdot 27

4 \cdot (20 + 7)

4 \cdot 20 + 4 \cdot 7

80 + 28

108

Δ = 49 + 108

ⓘ Start from

49

and move forward

108

on the number line:

49 + 108

157

Δ = 157

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 7 - \sqrt{157}}{2 \cdot 1}

x_{1} = \frac{- 7 - \sqrt{157}}{2}

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

- 7 - \sqrt{157}

- (7 + \sqrt{157})

x_{1} = - \frac{7 + \sqrt{157}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 7 + \sqrt{157}}{2 \cdot 1}

x_{2} = \frac{- 7 + \sqrt{157}}{2}

▶Solutions obtained:

x = {- \frac{7 + \sqrt{157}}{2}, \frac{\sqrt{157} - 7}{2}}

2) The numeric value of:

x = {- \frac{7 + \sqrt{157}}{2}, \frac{\sqrt{157} - 7}{2}}

equals:

x = {\approx - \frac{7 + 12.5300}{2}, \approx \frac{12.5300 - 7}{2}}

x = {\approx - \frac{19.5300}{2}, \approx \frac{5.5300}{2}}

▷

x = {\approx -9.7650, \approx 2.7650}