-x^2+3x=x^2+x-4 — step-by-step solution

Let's solve in

ℝ

- x^{2} + 3 x = x^{2} + x - 4

- x^{2}

+ 3 x

- x^{2}

- x

+ 4 = 0

ⓘ Like terms are grouped by adding their coefficients:

- x^{2} - x^{2}

-2 x^{2}

3 x - x

2 x

- 2 x^{2} + 2 x + 4 = 0

Since

-2 \neq 0

, divide both sides by

-2

x^{2} - x - 2 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -1

and

c = -2

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-2)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

Δ = 1 - 4 \cdot (-2)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-2)

8

Δ = 1 + 8

ⓘ Start from

1

and move forward

8

on the number line:

1 + 8

9

Δ = 9

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{9}}{2 \cdot 1}

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

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

\sqrt{9}

3

;

3^{2}

9

x_{1} = \frac{1 - 3}{2}

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

1 - 3

-2

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

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

x_{1} = -1

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{9}}{2 \cdot 1}

x_{2} = \frac{1 + \sqrt{9}}{2}

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

\sqrt{9}

3

;

3^{2}

9

x_{2} = \frac{1 + 3}{2}

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

1 + 3

4

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

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

2

, their greatest common divisor:

\frac{4}{2}

2

x_{2} = 2

▶Solutions found:

x = {-1, 2}