2x^2+3=5x+6 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} + 3 = 5 x + 6

2 x^{2}

+ 3

- 5 x

- 6 = 0

ⓘ Start from

3

and move back

6

on the number line, going below zero:

3 - 6

-3

2 x^{2} - 3 - 5 x = 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 = -5

and

c = -3

Δ = {(-5)}^{2} - 4 \cdot 2 \cdot (-3)

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

{(-5)}^{2}

(-5) \cdot (-5)

25

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Two minus signs cancel out, the product of two negative numbers is positive:

- 8 \cdot (-3)

8 \cdot 3

Δ = 25 + 8 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 3

24

Δ = 25 + 24

ⓘ Start from

25

and move forward

24

on the number line:

25 + 24

49

Δ = 49

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-5) - \sqrt{49}}{2 \cdot 2}

x_{1} = \frac{5 - \sqrt{49}}{2 \cdot 2}

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

\sqrt{49}

7

;

7^{2}

49

ⅰ

2 \cdot 2

4

x_{1} = \frac{5 - 7}{4}

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

5 - 7

-2

x_{1} = - \frac{2}{4}

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

2

, their greatest common divisor:

- \frac{2}{4}

- \frac{1 \cdot 2}{2 \cdot 2}

- \frac{1}{2}

x_{1} = - \frac{1}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-5) + \sqrt{49}}{2 \cdot 2}

x_{2} = \frac{5 + \sqrt{49}}{2 \cdot 2}

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

\sqrt{49}

7

;

7^{2}

49

ⅰ

2 \cdot 2

4

x_{2} = \frac{5 + 7}{4}

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

5 + 7

12

x_{2} = \frac{12}{4}

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

4

, their greatest common divisor:

\frac{12}{4}

3

x_{2} = 3

▶Solutions obtained:

x = {- \frac{1}{2}, 3}

2) The numeric value of:

x = {- \frac{1}{2}, 3}

equals:

▷

x = {-0.5, 3}