2x^2-12x+11=0 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} - 12 x + 11 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = -12

and

c = 11

Δ = {(-12)}^{2} - 4 \cdot 2 \cdot 11

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

{(-12)}^{2}

(-12) \cdot (-12)

144

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 144 - 8 \cdot 11

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 11

8 \cdot (10 + 1)

8 \cdot 10 + 8

80 + 8

88

Δ = 144 - 88

ⓘ Start from

144

and move back

88

on the number line:

144 - 88

56

Δ = 56

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-12) - \sqrt{56}}{2 \cdot 2}

x_{1} = \frac{12 - \sqrt{56}}{2 \cdot 2}

ⓘ We look for the largest square dividing

56

4

2^{2}

, and take it out of the root:

\sqrt{56}

\sqrt{4 \cdot 14}

\sqrt{4} \cdot \sqrt{14}

2 \cdot \sqrt{14}

ⅰ

2 \cdot 2

4

x_{1} = \frac{12 - 2 \cdot \sqrt{14}}{4}

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

\frac{12 - 2 \cdot \sqrt{14}}{4}

\frac{6 - \sqrt{14}}{2}

x_{1} = \frac{6 - \sqrt{14}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-12) + \sqrt{56}}{2 \cdot 2}

x_{2} = \frac{12 + \sqrt{56}}{2 \cdot 2}

ⓘ We look for the largest square dividing

56

4

2^{2}

, and take it out of the root:

\sqrt{56}

\sqrt{4 \cdot 14}

\sqrt{4} \cdot \sqrt{14}

2 \cdot \sqrt{14}

ⅰ

2 \cdot 2

4

x_{2} = \frac{12 + 2 \cdot \sqrt{14}}{4}

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

\frac{12 + 2 \cdot \sqrt{14}}{4}

\frac{6 + \sqrt{14}}{2}

x_{2} = \frac{6 + \sqrt{14}}{2}

▶Solutions obtained:

x = {\frac{6 - \sqrt{14}}{2}, \frac{6 + \sqrt{14}}{2}}

2) The numeric value of:

x = {\frac{6 - \sqrt{14}}{2}, \frac{6 + \sqrt{14}}{2}}

equals:

x = {\approx \frac{6 - 3.7417}{2}, \approx \frac{6 + 3.7417}{2}}

x = {\approx \frac{2.2583}{2}, \approx \frac{9.7417}{2}}

▷

x = {\approx 1.1292, \approx 4.8708}