5x^2+x-6=0 — step-by-step solution

1) Let's solve in

ℝ

5 x^{2} + x - 6 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 5

b = 1

and

c = -6

Δ = 1 - 4 \cdot 5 \cdot (-6)

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 5

20

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

- 20 \cdot (-6)

20 \cdot 6

Δ = 1 + 20 \cdot 6

ⓘ Multiplications are done before additions and subtractions:

20 \cdot 6

120

Δ = 1 + 120

ⓘ Start from

1

and move forward

120

on the number line:

1 + 120

121

Δ = 121

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{121}}{2 \cdot 5}

x_{1} = \frac{- 1 - \sqrt{121}}{2 \cdot 5}

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

\sqrt{121}

11

;

11^{2}

121

ⅰ

2 \cdot 5

10

x_{1} = \frac{- 1 - 11}{10}

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

- 1 - 11

-12

x_{1} = - \frac{12}{10}

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

2

, their greatest common divisor:

- \frac{12}{10}

- \frac{6 \cdot 2}{5 \cdot 2}

- \frac{6}{5}

x_{1} = - \frac{6}{5}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{121}}{2 \cdot 5}

x_{2} = \frac{- 1 + \sqrt{121}}{2 \cdot 5}

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

\sqrt{121}

11

;

11^{2}

121

ⅰ

2 \cdot 5

10

x_{2} = \frac{- 1 + 11}{10}

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

- 1 + 11

10

x_{2} = \frac{10}{10}

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

10

, their greatest common divisor:

\frac{10}{10}

1

x_{2} = 1

▶Solutions found:

x = {- \frac{6}{5}, 1}

2) The numeric value of:

x = {- \frac{6}{5}, 1}

equals:

▷

x = {-1.2, 1}