89x^2+38x-45=0 — step-by-step solution

1) Let's solve in

ℝ

89 x^{2} + 38 x - 45 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 89

b = 38

and

c = -45

Δ = 38^{2} - 4 \cdot 89 \cdot (-45)

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

38^{2}

38 \cdot 38

1444

■Multiplications are done before additions and subtractions:

4 \cdot 89

4 \cdot (80 + 9)

4 \cdot 80 + 4 \cdot 9

320 + 36

356

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

- 356 \cdot (-45)

356 \cdot 45

Δ = 1444 + 356 \cdot 45

ⓘ Multiplications are done before additions and subtractions:

356 \cdot 45

356 \cdot (40 + 5)

356 \cdot 40 + 356 \cdot 5

14,240 + 1780

16,020

Δ = 1444 + 16,020

ⓘ Start from

1444

and move forward

16,020

on the number line:

1444 + 16,020

17,464

Δ = 17,464

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 38 - \sqrt{17,464}}{2 \cdot 89}

x_{1} = \frac{- 38 - \sqrt{17,464}}{2 \cdot 89}

ⓘ We look for the largest square dividing

17,464

4

2^{2}

, and take it out of the root:

\sqrt{17,464}

\sqrt{4 \cdot 4366}

\sqrt{4} \cdot \sqrt{4366}

2 \cdot \sqrt{4366}

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

- 38 - 2 \cdot \sqrt{4366}

- (38 + 2 \cdot \sqrt{4366})

ⅰ

2 \cdot 89

2 \cdot (80 + 9)

2 \cdot 80 + 2 \cdot 9

160 + 18

178

x_{1} = - \frac{38 + 2 \cdot \sqrt{4366}}{178}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{38 + 2 \cdot \sqrt{4366}}{178}

- \frac{19 + \sqrt{4366}}{89}

x_{1} = - \frac{19 + \sqrt{4366}}{89}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 38 + \sqrt{17,464}}{2 \cdot 89}

x_{2} = \frac{- 38 + \sqrt{17,464}}{2 \cdot 89}

ⓘ We look for the largest square dividing

17,464

4

2^{2}

, and take it out of the root:

\sqrt{17,464}

\sqrt{4 \cdot 4366}

\sqrt{4} \cdot \sqrt{4366}

2 \cdot \sqrt{4366}

ⅰ

2 \cdot 89

2 \cdot (80 + 9)

2 \cdot 80 + 2 \cdot 9

160 + 18

178

x_{2} = \frac{- 38 + 2 \cdot \sqrt{4366}}{178}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{- 38 + 2 \cdot \sqrt{4366}}{178}

\frac{- 19 + \sqrt{4366}}{89}

x_{2} = \frac{- 19 + \sqrt{4366}}{89}

▶Solutions obtained:

x = {- \frac{19 + \sqrt{4366}}{89}, \frac{\sqrt{4366} - 19}{89}}

2) The numeric value of:

x = {- \frac{19 + \sqrt{4366}}{89}, \frac{\sqrt{4366} - 19}{89}}

equals:

x = {\approx - \frac{19 + 66.0757}{89}, \approx \frac{66.0757 - 19}{89}}

x = {\approx - \frac{85.0757}{89}, \approx \frac{47.0757}{89}}

▷

x = {\approx -0.9559, \approx 0.5289}