x^2-12x+3<0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - 12 x + 3 < 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 = -12

and

c = 3

Δ = {(-12)}^{2} - 4 \cdot 1 \cdot 3

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

{(-12)}^{2}

(-12) \cdot (-12)

144

Δ = 144 - 4 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

Δ = 144 - 12

ⓘ Start from

144

and move back

12

on the number line:

144 - 12

132

Δ = 132

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-12) - \sqrt{132}}{2 \cdot 1}

x_{1} = \frac{12 - \sqrt{132}}{2}

ⓘ We look for the largest square dividing

132

4

2^{2}

, and take it out of the root:

\sqrt{132}

\sqrt{4 \cdot 33}

\sqrt{4} \cdot \sqrt{33}

2 \cdot \sqrt{33}

x_{1} = \frac{12 - 2 \cdot \sqrt{33}}{2}

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

\frac{12 - 2 \cdot \sqrt{33}}{2}

\frac{2 \cdot (6 - \sqrt{33})}{2}

6 - \sqrt{33}

x_{1} = 6 - \sqrt{33}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-12) + \sqrt{132}}{2 \cdot 1}

x_{2} = \frac{12 + \sqrt{132}}{2}

ⓘ We look for the largest square dividing

132

4

2^{2}

, and take it out of the root:

\sqrt{132}

\sqrt{4 \cdot 33}

\sqrt{4} \cdot \sqrt{33}

2 \cdot \sqrt{33}

x_{2} = \frac{12 + 2 \cdot \sqrt{33}}{2}

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

\frac{12 + 2 \cdot \sqrt{33}}{2}

\frac{2 \cdot (6 + \sqrt{33})}{2}

6 + \sqrt{33}

x_{2} = 6 + \sqrt{33}

The quadratic has the opposite sign to a =

1

between its roots, so it is indeed negative between its roots; the roots, where it is zero, are excluded.

▶Solution obtained:

x \in (6 - \sqrt{33}, 6 + \sqrt{33})

2) The numeric value of:

x \in (6 - \sqrt{33}, 6 + \sqrt{33})

equals:

x \in (6 - \approx 5.7446, 6 + \approx 5.7446)

▷

x \in (\approx 0.2554, \approx 11.7446)