x^2=13+6x — step-by-step solution

1) Let's solve in

ℝ

x^{2} = 13 + 6 x

x^{2} - 6 x - 13 = 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 = -6

and

c = -13

Δ = {(-6)}^{2} - 4 \cdot 1 \cdot (-13)

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

{(-6)}^{2}

(-6) \cdot (-6)

36

Δ = 36 - 4 \cdot (-13)

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

- 4 \cdot (-13)

4 \cdot 13

4 \cdot (10 + 3)

4 \cdot 10 + 4 \cdot 3

40 + 12

52

Δ = 36 + 52

ⓘ Start from

36

and move forward

52

on the number line:

36 + 52

88

Δ = 88

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-6) - \sqrt{88}}{2 \cdot 1}

x_{1} = \frac{6 - \sqrt{88}}{2}

ⓘ We look for the largest square dividing

88

4

2^{2}

, and take it out of the root:

\sqrt{88}

\sqrt{4 \cdot 22}

\sqrt{4} \cdot \sqrt{22}

2 \cdot \sqrt{22}

x_{1} = \frac{6 - 2 \cdot \sqrt{22}}{2}

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

\frac{6 - 2 \cdot \sqrt{22}}{2}

\frac{2 \cdot (3 - \sqrt{22})}{2}

3 - \sqrt{22}

x_{1} = 3 - \sqrt{22}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-6) + \sqrt{88}}{2 \cdot 1}

x_{2} = \frac{6 + \sqrt{88}}{2}

ⓘ We look for the largest square dividing

88

4

2^{2}

, and take it out of the root:

\sqrt{88}

\sqrt{4 \cdot 22}

\sqrt{4} \cdot \sqrt{22}

2 \cdot \sqrt{22}

x_{2} = \frac{6 + 2 \cdot \sqrt{22}}{2}

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

\frac{6 + 2 \cdot \sqrt{22}}{2}

\frac{2 \cdot (3 + \sqrt{22})}{2}

3 + \sqrt{22}

x_{2} = 3 + \sqrt{22}

▶Solutions obtained:

x = {3 - \sqrt{22}, 3 + \sqrt{22}}

2) The numeric value of:

x = {3 - \sqrt{22}, 3 + \sqrt{22}}

is:

x = {\approx 3 - 4.6904, \approx 3 + 4.6904}

▷

x = {\approx -1.6904, \approx 7.6904}