199-2x^2=x^2+33x+28 — step-by-step solution

1) Let's solve in

ℝ

199 - 2 x^{2} = x^{2} + 33 x + 28

199

- 2 x^{2}

- x^{2}

- 33 x

- 28 = 0

ⓘ Start from

199

and move back

28

on the number line:

199 - 28

171

■Like terms are grouped by adding their coefficients:

- 2 x^{2} - x^{2}

-3 x^{2}

171 - 3 x^{2} - 33 x = 0

Since

-3 \neq 0

, divide both sides by

-3

- 57 + x^{2} + 11 x = 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 = 11

and

c = -57

Δ = 11^{2} - 4 \cdot 1 \cdot (-57)

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

11^{2}

11 \cdot 11

121

Δ = 121 - 4 \cdot (-57)

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

- 4 \cdot (-57)

4 \cdot 57

4 \cdot (50 + 7)

4 \cdot 50 + 4 \cdot 7

200 + 28

228

Δ = 121 + 228

ⓘ Start from

121

and move forward

228

on the number line:

121 + 228

349

Δ = 349

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 11 - \sqrt{349}}{2 \cdot 1}

x_{1} = \frac{- 11 - \sqrt{349}}{2}

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

- 11 - \sqrt{349}

- (11 + \sqrt{349})

x_{1} = - \frac{11 + \sqrt{349}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 11 + \sqrt{349}}{2 \cdot 1}

x_{2} = \frac{- 11 + \sqrt{349}}{2}

▶Solutions obtained:

x = {- \frac{11 + \sqrt{349}}{2}, \frac{\sqrt{349} - 11}{2}}

2) The numeric value of:

x = {- \frac{11 + \sqrt{349}}{2}, \frac{\sqrt{349} - 11}{2}}

is:

x = {\approx - \frac{11 + 18.6815}{2}, \approx \frac{18.6815 - 11}{2}}

x = {\approx - \frac{29.6815}{2}, \approx \frac{7.6815}{2}}

▷

x = {\approx -14.8408, \approx 3.8408}