2x^2-9x+11=4 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} - 9 x + 11 = 4

ⓘ We subtract

4

from both sides

2 x^{2}

- 9 x

+ 11

- 4 = 0

ⓘ Start from

11

and move back

4

on the number line:

11 - 4

7

2 x^{2} - 9 x + 7 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = -9

and

c = 7

Δ = {(-9)}^{2} - 4 \cdot 2 \cdot 7

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

{(-9)}^{2}

(-9) \cdot (-9)

81

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 81 - 8 \cdot 7

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 7

56

Δ = 81 - 56

ⓘ Start from

81

and move back

56

on the number line:

81 - 56

25

Δ = 25

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-9) - \sqrt{25}}{2 \cdot 2}

x_{1} = \frac{9 - \sqrt{25}}{2 \cdot 2}

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

\sqrt{25}

5

;

5^{2}

25

ⅰ

2 \cdot 2

4

x_{1} = \frac{9 - 5}{4}

ⓘ The numerator and the denominator are computed before the division:

9 - 5

4

x_{1} = \frac{4}{4}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

\frac{4}{4}

1

x_{1} = 1

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-9) + \sqrt{25}}{2 \cdot 2}

x_{2} = \frac{9 + \sqrt{25}}{2 \cdot 2}

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

\sqrt{25}

5

;

5^{2}

25

ⅰ

2 \cdot 2

4

x_{2} = \frac{9 + 5}{4}

ⓘ The numerator and the denominator are computed before the division:

9 + 5

14

x_{2} = \frac{14}{4}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{14}{4}

\frac{7 \cdot 2}{2 \cdot 2}

\frac{7}{2}

x_{2} = \frac{7}{2}

▶Solutions obtained:

x = {1, \frac{7}{2}}

2) The numeric value of:

x = {1, \frac{7}{2}}

is:

▷

x = {1, 3.5}