x^2+(-(3x)/7)^2=16 — step-by-step solution

1) Let's solve in

ℝ

x^{2} + {(- \frac{3 x}{7})}^{2} = 16

ⓘ Powers are computed before multiplications, additions and subtractions. With an even integer exponent, the minus sign in the base disappears; with an odd integer exponent, it moves in front of the power:

{(- \frac{3 x}{7})}^{2}

{(\frac{3 x}{7})}^{2}

x^{2} + {(\frac{3 x}{7})}^{2} = 16

ⓘ We subtract

16

from both sides

x^{2} + {(\frac{3 x}{7})}^{2} - 16 = 0

ⓘ The power of a quotient is the quotient of the powers, (a/b)^n = a^n/b^n:

{(\frac{3 x}{7})}^{2}

\frac{{(3 x)}^{2}}{7^{2}}

x^{2} + \frac{{(3 x)}^{2}}{7^{2}} - 16 = 0

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

7^{2}

7 \cdot 7

49

x^{2} + \frac{{(3 x)}^{2}}{49} - 16 = 0

ⓘ The power of a product is the product of the powers:

{(3 x)}^{2}

3^{2} \cdot x^{2}

x^{2} + \frac{3^{2} \cdot x^{2}}{49} - 16 = 0

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

3^{2}

3 \cdot 3

9

x^{2} + \frac{9 x^{2}}{49} - 16 = 0

ⓘ Like terms are grouped by adding their coefficients:

x^{2} + \frac{9 x^{2}}{49}

\frac{58 x^{2}}{49}

\frac{58 x^{2}}{49} - 16 = 0

Since

2 \neq 0

, divide both sides by

2

\frac{29 x^{2}}{49} - 8 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 29 \cdot \frac{1}{49}

b = 0

and

c = -8

Δ = 0^{2} - 4 \cdot 29 \cdot \frac{1}{49} \cdot (-8)

ⓘ Adding 0 changes nothing:

0 - 4 \cdot 29 \cdot \frac{1}{49} \cdot (-8)

- 4 \cdot 29 \cdot \frac{1}{49} \cdot (-8)

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

-4 \cdot (-8)

4 \cdot 8

Δ = 4 \cdot 29 \cdot \frac{1}{49} \cdot 8

ⓘ Multiplying a fraction by a number multiplies its numerator:

116 \cdot \frac{1}{49} \cdot 8

\frac{116 \cdot 1 \cdot 8}{49}

\frac{116 \cdot 8}{49}

ⅰ

4 \cdot 29

4 \cdot (20 + 9)

4 \cdot 20 + 4 \cdot 9

80 + 36

116

Δ = \frac{116 \cdot 8}{49}

ⅰ

116 \cdot 8

(100 + 10 + 6) \cdot 8

100 \cdot 8 + 10 \cdot 8 + 6 \cdot 8

800 + 80 + 48

928

Δ = \frac{928}{49}

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 0 - \sqrt{\frac{928}{49}}}{2 \cdot 29 \cdot \frac{1}{49}}

x_{1} = - \frac{\sqrt{\frac{928}{49}}}{2 \cdot 29 \cdot \frac{1}{49}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

58 \cdot \frac{1}{49}

\frac{58 \cdot 1}{49}

\frac{58}{49}

ⅰ

2 \cdot 29

2 \cdot (20 + 9)

2 \cdot 20 + 2 \cdot 9

40 + 18

58

x_{1} = - \frac{\sqrt{\frac{928}{49}}}{\frac{58}{49}}

ⓘ Dividing is multiplying by the reciprocal:

- \frac{\sqrt{\frac{928}{49}}}{\frac{58}{49}}

- \sqrt{\frac{928}{49}} \cdot \frac{49}{58}

x_{1} = - \sqrt{\frac{928}{49}} \cdot \frac{49}{58}

ⓘ Multiplying a fraction by a term multiplies its numerator:

- \sqrt{\frac{928}{49}} \cdot \frac{49}{58}

- \frac{\sqrt{\frac{928}{49}} \cdot 49}{58}

x_{1} = - \frac{\sqrt{\frac{928}{49}} \cdot 49}{58}

ⓘ The root of a quotient is the quotient of the roots, sqrt(a/b) = sqrt(a)/sqrt(b):

\sqrt{\frac{928}{49}}

\frac{\sqrt{928}}{\sqrt{49}}

x_{1} = - \frac{\frac{\sqrt{928}}{\sqrt{49}} \cdot 49}{58}

ⓘ We look for the largest square dividing

928

16

4^{2}

, and take it out of the root:

\sqrt{928}

\sqrt{16 \cdot 58}

\sqrt{16} \cdot \sqrt{58}

4 \cdot \sqrt{58}

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

\sqrt{49}

7

;

7^{2}

49

x_{1} = - \frac{\frac{4 \cdot \sqrt{58}}{7} \cdot 49}{58}

ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{4 \cdot \sqrt{58}}{7} \cdot 49

\frac{4 \cdot \sqrt{58} \cdot 49}{7}

\frac{196 \cdot \sqrt{58}}{7}

x_{1} = - \frac{\frac{196 \cdot \sqrt{58}}{7}}{58}

ⓘ Dividing a fraction by

58

is multiplying its denominator by

58

- \frac{\frac{196 \cdot \sqrt{58}}{7}}{58}

- \frac{196 \cdot \sqrt{58}}{7 \cdot 58}

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

- \frac{196 \cdot \sqrt{58}}{7 \cdot 58}

- \frac{28 \cdot \sqrt{58}}{58}

x_{1} = - \frac{28 \cdot \sqrt{58}}{58}

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

- \frac{28 \cdot \sqrt{58}}{58}

- \frac{14 \cdot \sqrt{58}}{29}

x_{1} = - \frac{14 \cdot \sqrt{58}}{29}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 0 + \sqrt{\frac{928}{49}}}{2 \cdot 29 \cdot \frac{1}{49}}

x_{2} = \frac{\sqrt{\frac{928}{49}}}{2 \cdot 29 \cdot \frac{1}{49}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

58 \cdot \frac{1}{49}

\frac{58 \cdot 1}{49}

\frac{58}{49}

ⅰ

2 \cdot 29

2 \cdot (20 + 9)

2 \cdot 20 + 2 \cdot 9

40 + 18

58

x_{2} = \frac{\sqrt{\frac{928}{49}}}{\frac{58}{49}}

ⓘ Dividing is multiplying by the reciprocal:

\frac{\sqrt{\frac{928}{49}}}{\frac{58}{49}}

\sqrt{\frac{928}{49}} \cdot \frac{49}{58}

x_{2} = \sqrt{\frac{928}{49}} \cdot \frac{49}{58}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\sqrt{\frac{928}{49}} \cdot \frac{49}{58}

\frac{\sqrt{\frac{928}{49}} \cdot 49}{58}

x_{2} = \frac{\sqrt{\frac{928}{49}} \cdot 49}{58}

ⓘ The root of a quotient is the quotient of the roots, sqrt(a/b) = sqrt(a)/sqrt(b):

\sqrt{\frac{928}{49}}

\frac{\sqrt{928}}{\sqrt{49}}

x_{2} = \frac{\frac{\sqrt{928}}{\sqrt{49}} \cdot 49}{58}

ⓘ We look for the largest square dividing

928

16

4^{2}

, and take it out of the root:

\sqrt{928}

\sqrt{16 \cdot 58}

\sqrt{16} \cdot \sqrt{58}

4 \cdot \sqrt{58}

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

\sqrt{49}

7

;

7^{2}

49

x_{2} = \frac{\frac{4 \cdot \sqrt{58}}{7} \cdot 49}{58}

ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{4 \cdot \sqrt{58}}{7} \cdot 49

\frac{4 \cdot \sqrt{58} \cdot 49}{7}

\frac{196 \cdot \sqrt{58}}{7}

x_{2} = \frac{\frac{196 \cdot \sqrt{58}}{7}}{58}

ⓘ Dividing a fraction by

58

is multiplying its denominator by

58

\frac{\frac{196 \cdot \sqrt{58}}{7}}{58}

\frac{196 \cdot \sqrt{58}}{7 \cdot 58}

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

\frac{196 \cdot \sqrt{58}}{7 \cdot 58}

\frac{28 \cdot \sqrt{58}}{58}

x_{2} = \frac{28 \cdot \sqrt{58}}{58}

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

\frac{28 \cdot \sqrt{58}}{58}

\frac{14 \cdot \sqrt{58}}{29}

x_{2} = \frac{14 \cdot \sqrt{58}}{29}

▶Solutions obtained:

x = {- \frac{14 \cdot \sqrt{58}}{29}, \frac{14 \cdot \sqrt{58}}{29}}

2) The numeric value of:

x = {- \frac{14 \cdot \sqrt{58}}{29}, \frac{14 \cdot \sqrt{58}}{29}}

is:

x = {\approx - \frac{14 \cdot 7.6158}{29}, \approx \frac{14 \cdot 7.6158}{29}}

x = {\approx - \frac{106.6208}{29}, \approx \frac{106.6208}{29}}

▷

x = {\approx -3.6766, \approx 3.6766}