a-4=16-a^2 — step-by-step solution

Let's solve in

ℝ

a - 4 = 16 - a^{2}

a

- 4

- 16

+ a^{2} = 0

ⓘ Start from

-4

and move back

16

more on the number line, away from zero:

- 4 - 16

-20

a - 20 + a^{2} = 0

Given the quadratic form

A a^{2} + B a + C

, you can calculate the discriminant:

Δ = B^{2} - 4 A C

. Here

A = 1

B = 1

and

C = -20

Δ = 1 - 4 \cdot 1 \cdot (-20)

Δ = 1 - 4 \cdot (-20)

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

- 4 \cdot (-20)

80

Δ = 1 + 80

ⓘ Start from

1

and move forward

80

on the number line:

1 + 80

81

Δ = 81

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

a_{1}

\frac{- B - \sqrt{Δ}}{2 A}

\frac{- 1 - \sqrt{81}}{2 \cdot 1}

a_{1} = \frac{- 1 - \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

a_{1} = \frac{- 1 - 9}{2}

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

- 1 - 9

-10

a_{1} = - \frac{10}{2}

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

2

, their greatest common divisor:

- \frac{10}{2}

-5

a_{1} = -5

The quadratic formula gives

a_{2}

\frac{- B + \sqrt{Δ}}{2 A}

\frac{- 1 + \sqrt{81}}{2 \cdot 1}

a_{2} = \frac{- 1 + \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

a_{2} = \frac{- 1 + 9}{2}

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

- 1 + 9

8

a_{2} = \frac{8}{2}

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

2

, their greatest common divisor:

\frac{8}{2}

4

a_{2} = 4

▶Solutions found:

a = {-5, 4}