-4g^2-68g=172 — step-by-step solution

1) Let's solve in

ℝ

- 4 g^{2} - 68 g = 172

Since

-4 \neq 0

, divide both sides by

-4

g^{2} + 17 g = -43

g^{2} + 17 g + 43 = 0

Given the quadratic form

a g^{2} + b g + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 17

and

c = 43

Δ = 17^{2} - 4 \cdot 1 \cdot 43

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

17^{2}

17 \cdot 17

289

Δ = 289 - 4 \cdot 43

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 43

4 \cdot (40 + 3)

4 \cdot 40 + 4 \cdot 3

160 + 12

172

Δ = 289 - 172

ⓘ Start from

289

and move back

172

on the number line:

289 - 172

117

Δ = 117

Δ > 0

, the equation has two roots:

The quadratic formula gives

g_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 17 - \sqrt{117}}{2 \cdot 1}

g_{1} = \frac{- 17 - \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:

- 17 - 3 \cdot \sqrt{13}

- (17 + 3 \cdot \sqrt{13})

g_{1} = - \frac{17 + 3 \cdot \sqrt{13}}{2}

The quadratic formula gives

g_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 17 + \sqrt{117}}{2 \cdot 1}

g_{2} = \frac{- 17 + \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}

g_{2} = \frac{- 17 + 3 \cdot \sqrt{13}}{2}

▶Solutions obtained:

g = {- \frac{17 + 3 \cdot \sqrt{13}}{2}, \frac{3 \cdot \sqrt{13} - 17}{2}}

2) The numeric value of:

g = {- \frac{17 + 3 \cdot \sqrt{13}}{2}, \frac{3 \cdot \sqrt{13} - 17}{2}}

is:

g = {\approx - \frac{17 + 3 \cdot 3.6056}{2}, \approx \frac{3 \cdot 3.6056 - 17}{2}}

g = {\approx - \frac{17 + 10.8167}{2}, \approx \frac{10.8167 - 17}{2}}

g = {\approx - \frac{27.8167}{2}, \approx - \frac{6.1833}{2}}

▷

g = {\approx -13.9083, \approx -3.0917}