y-5=y^2-y-9 — step-by-step solution

1) Let's solve in

ℝ

y - 5 = y^{2} - y - 9

y

- 5

- y^{2}

+ y

+ 9 = 0

ⓘ Start from

-5

and move forward

9

on the number line, going past zero:

- 5 + 9

4

■Like terms are grouped by adding their coefficients:

y + y

2 y

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

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

Since

-1 \neq 0

, divide both sides by

-1

y^{2} - 2 y - 4 = 0

Given the quadratic form

a y^{2} + b y + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -2

and

c = -4

Δ = {(-2)}^{2} - 4 \cdot 1 \cdot (-4)

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

{(-2)}^{2}

(-2) \cdot (-2)

4

Δ = 4 - 4 \cdot (-4)

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

- 4 \cdot (-4)

16

Δ = 4 + 16

ⓘ Start from

4

and move forward

16

on the number line:

4 + 16

20

Δ = 20

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

y_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-2) - \sqrt{20}}{2 \cdot 1}

y_{1} = \frac{2 - \sqrt{20}}{2}

ⓘ We look for the largest square dividing

20

4

2^{2}

, and take it out of the root:

\sqrt{20}

\sqrt{4 \cdot 5}

\sqrt{4} \cdot \sqrt{5}

2 \cdot \sqrt{5}

y_{1} = \frac{2 - 2 \cdot \sqrt{5}}{2}

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

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

\frac{2 \cdot (1 - \sqrt{5})}{2}

1 - \sqrt{5}

y_{1} = 1 - \sqrt{5}

The quadratic formula gives

y_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-2) + \sqrt{20}}{2 \cdot 1}

y_{2} = \frac{2 + \sqrt{20}}{2}

ⓘ We look for the largest square dividing

20

4

2^{2}

, and take it out of the root:

\sqrt{20}

\sqrt{4 \cdot 5}

\sqrt{4} \cdot \sqrt{5}

2 \cdot \sqrt{5}

y_{2} = \frac{2 + 2 \cdot \sqrt{5}}{2}

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

\frac{2 + 2 \cdot \sqrt{5}}{2}

\frac{2 \cdot (1 + \sqrt{5})}{2}

1 + \sqrt{5}

y_{2} = 1 + \sqrt{5}

▶Solutions obtained:

y = {1 - \sqrt{5}, 1 + \sqrt{5}}

2) The numeric value of:

y = {1 - \sqrt{5}, 1 + \sqrt{5}}

is:

y = {\approx 1 - 2.2361, \approx 1 + 2.2361}

▷

y = {\approx -1.2361, \approx 3.2361}