r^2-13r+42=0 — step-by-step solution

Let's solve in

ℝ

r^{2} - 13 r + 42 = 0

Given the quadratic form

a r^{2} + b r + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -13

and

c = 42

Δ = {(-13)}^{2} - 4 \cdot 1 \cdot 42

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

{(-13)}^{2}

(-13) \cdot (-13)

169

Δ = 169 - 4 \cdot 42

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 42

4 \cdot (40 + 2)

4 \cdot 40 + 4 \cdot 2

160 + 8

168

Δ = 169 - 168

ⓘ Start from

169

and move back

168

on the number line:

169 - 168

1

Δ = 1

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

r_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-13) - \sqrt{1}}{2 \cdot 1}

r_{1} = \frac{13 - 1}{2}

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

13 - 1

12

r_{1} = \frac{12}{2}

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

2

, their greatest common divisor:

\frac{12}{2}

6

r_{1} = 6

The quadratic formula gives

r_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-13) + \sqrt{1}}{2 \cdot 1}

r_{2} = \frac{13 + 1}{2}

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

13 + 1

14

r_{2} = \frac{14}{2}

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

2

, their greatest common divisor:

\frac{14}{2}

7

r_{2} = 7

▶Solutions obtained:

r = {6, 7}