5x^2-35x+60=0 — step-by-step solution

Let's solve in

ℝ

5 x^{2} - 35 x + 60 = 0

Since

5 \neq 0

, divide both sides by

5

x^{2} - 7 x + 12 = 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 = -7

and

c = 12

Δ = {(-7)}^{2} - 4 \cdot 1 \cdot 12

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

{(-7)}^{2}

(-7) \cdot (-7)

49

Δ = 49 - 4 \cdot 12

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

Δ = 49 - 48

ⓘ Start from

49

and move back

48

on the number line:

49 - 48

1

Δ = 1

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-7) - \sqrt{1}}{2 \cdot 1}

x_{1} = \frac{7 - 1}{2}

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

7 - 1

6

x_{1} = \frac{6}{2}

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

2

, their greatest common divisor:

\frac{6}{2}

3

x_{1} = 3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-7) + \sqrt{1}}{2 \cdot 1}

x_{2} = \frac{7 + 1}{2}

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

7 + 1

8

x_{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

x_{2} = 4

▶Solutions found:

x = {3, 4}