x^2-14x+36=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - 14 x + 36 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -14

and

c = 36

Δ = {(-14)}^{2} - 4 \cdot 1 \cdot 36

ⓘ 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 36

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 36

4 \cdot (30 + 6)

4 \cdot 30 + 4 \cdot 6

120 + 24

144

Δ = 196 - 144

ⓘ Start from

196

and move back

144

on the number line:

196 - 144

52

Δ = 52

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-14) - \sqrt{52}}{2 \cdot 1}

x_{1} = \frac{14 - \sqrt{52}}{2}

ⓘ We look for the largest square dividing

52

4

2^{2}

, and take it out of the root:

\sqrt{52}

\sqrt{4 \cdot 13}

\sqrt{4} \cdot \sqrt{13}

2 \cdot \sqrt{13}

x_{1} = \frac{14 - 2 \cdot \sqrt{13}}{2}

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

\frac{14 - 2 \cdot \sqrt{13}}{2}

\frac{2 \cdot (7 - \sqrt{13})}{2}

7 - \sqrt{13}

x_{1} = 7 - \sqrt{13}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-14) + \sqrt{52}}{2 \cdot 1}

x_{2} = \frac{14 + \sqrt{52}}{2}

ⓘ We look for the largest square dividing

52

4

2^{2}

, and take it out of the root:

\sqrt{52}

\sqrt{4 \cdot 13}

\sqrt{4} \cdot \sqrt{13}

2 \cdot \sqrt{13}

x_{2} = \frac{14 + 2 \cdot \sqrt{13}}{2}

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

\frac{14 + 2 \cdot \sqrt{13}}{2}

\frac{2 \cdot (7 + \sqrt{13})}{2}

7 + \sqrt{13}

x_{2} = 7 + \sqrt{13}

▶Solutions obtained:

x = {7 - \sqrt{13}, 7 + \sqrt{13}}

2) The numeric value of:

x = {7 - \sqrt{13}, 7 + \sqrt{13}}

equals:

x = {\approx 7 - 3.6056, \approx 7 + 3.6056}

▷

x = {\approx 3.3944, \approx 10.6056}