(36+x^2)/2 =7x-6 — step-by-step solution

1) Let's solve in

ℝ

\frac{36 + x^{2}}{2} = 7 x - 6

2) Let's add fractions:

\frac{36 + x^{2}}{2} - 7 x + 6 = 0

ⅰ

\frac{36 + x^{2}}{2} + 6

\frac{(36 + x^{2}) + 2 \cdot 6}{2}

\frac{(36 + x^{2}) + 2 \cdot 6}{2} - 7 x = 0

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 6

12

■To add fractions, put them over the same denominator, then add the numerators:

\frac{36 + x^{2} + 12}{2} - 7 x

\frac{(36 + x^{2} + 12) - 7 x \cdot 2}{2}

\frac{(36 + x^{2} + 12) - 7 x \cdot 2}{2} = 0

ⓘ The numerator and the denominator are computed before the division:

36 + 12

48

■Multiplications are done before additions and subtractions:

7 \cdot 2

14

\frac{48 + x^{2} - 14 x}{2} = 0

Let's solve in

ℝ

48 + x^{2} - 14 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 = -14

and

c = 48

Δ = {(-14)}^{2} - 4 \cdot 1 \cdot 48

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

{(-14)}^{2}

(-14) \cdot (-14)

196

Δ = 196 - 4 \cdot 48

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 48

4 \cdot (40 + 8)

4 \cdot 40 + 4 \cdot 8

160 + 32

192

Δ = 196 - 192

ⓘ Start from

196

and move back

192

on the number line:

196 - 192

4

Δ = 4

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-14) - \sqrt{4}}{2 \cdot 1}

x_{1} = \frac{14 - \sqrt{4}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

x_{1} = \frac{14 - 2}{2}

ⓘ The numerator and the denominator are computed before the division:

14 - 2

12

x_{1} = \frac{12}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{12}{2}

6

x_{1} = 6

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-14) + \sqrt{4}}{2 \cdot 1}

x_{2} = \frac{14 + \sqrt{4}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

x_{2} = \frac{14 + 2}{2}

ⓘ The numerator and the denominator are computed before the division:

14 + 2

16

x_{2} = \frac{16}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{16}{2}

8

x_{2} = 8

▶Solutions obtained:

x = {6, 8}