8x^2+9x=8 — step-by-step solution

1) Let's solve in

ℝ

8 x^{2} + 9 x = 8

8 x^{2} + 9 x - 8 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 8

b = 9

and

c = -8

Δ = 9^{2} - 4 \cdot 8 \cdot (-8)

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

9^{2}

9 \cdot 9

81

■Multiplications are done before additions and subtractions:

4 \cdot 8

32

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

- 32 \cdot (-8)

32 \cdot 8

Δ = 81 + 32 \cdot 8

ⓘ Multiplications are done before additions and subtractions:

32 \cdot 8

(30 + 2) \cdot 8

30 \cdot 8 + 2 \cdot 8

240 + 16

256

Δ = 81 + 256

ⓘ Start from

81

and move forward

256

on the number line:

81 + 256

337

Δ = 337

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 9 - \sqrt{337}}{2 \cdot 8}

x_{1} = \frac{- 9 - \sqrt{337}}{2 \cdot 8}

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

- 9 - \sqrt{337}

- (9 + \sqrt{337})

ⅰ

2 \cdot 8

16

x_{1} = - \frac{9 + \sqrt{337}}{16}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 9 + \sqrt{337}}{2 \cdot 8}

x_{2} = \frac{- 9 + \sqrt{337}}{2 \cdot 8}

ⅰ

2 \cdot 8

16

x_{2} = \frac{- 9 + \sqrt{337}}{16}

▶Solutions obtained:

x = {- \frac{9 + \sqrt{337}}{16}, \frac{\sqrt{337} - 9}{16}}

2) The numeric value of:

x = {- \frac{9 + \sqrt{337}}{16}, \frac{\sqrt{337} - 9}{16}}

is:

x = {\approx - \frac{9 + 18.3576}{16}, \approx \frac{18.3576 - 9}{16}}

x = {\approx - \frac{27.3576}{16}, \approx \frac{9.3576}{16}}

▷

x = {\approx -1.7098, \approx 0.5848}