15x^2+2x-10=0 — step-by-step solution

1) Let's solve in

ℝ

15 x^{2} + 2 x - 10 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 15

b = 2

and

c = -10

Δ = 2^{2} - 4 \cdot 15 \cdot (-10)

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

2^{2}

2 \cdot 2

4

■Multiplications are done before additions and subtractions:

4 \cdot 15

4 \cdot (10 + 5)

4 \cdot 10 + 4 \cdot 5

40 + 20

60

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

- 60 \cdot (-10)

60 \cdot 10

Δ = 4 + 60 \cdot 10

ⓘ Multiplications are done before additions and subtractions:

60 \cdot 10

600

Δ = 4 + 600

ⓘ Start from

4

and move forward

600

on the number line:

4 + 600

604

Δ = 604

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 2 - \sqrt{604}}{2 \cdot 15}

x_{1} = \frac{- 2 - \sqrt{604}}{2 \cdot 15}

ⓘ We look for the largest square dividing

604

4

2^{2}

, and take it out of the root:

\sqrt{604}

\sqrt{4 \cdot 151}

\sqrt{4} \cdot \sqrt{151}

2 \cdot \sqrt{151}

■The minus sign is factored out, changing the sign of each term:

- 2 - 2 \cdot \sqrt{151}

- (2 + 2 \cdot \sqrt{151})

ⅰ

2 \cdot 15

2 \cdot (10 + 5)

2 \cdot 10 + 2 \cdot 5

20 + 10

30

x_{1} = - \frac{2 + 2 \cdot \sqrt{151}}{30}

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

- \frac{2 + 2 \cdot \sqrt{151}}{30}

- \frac{1 + \sqrt{151}}{15}

x_{1} = - \frac{1 + \sqrt{151}}{15}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 2 + \sqrt{604}}{2 \cdot 15}

x_{2} = \frac{- 2 + \sqrt{604}}{2 \cdot 15}

ⓘ We look for the largest square dividing

604

4

2^{2}

, and take it out of the root:

\sqrt{604}

\sqrt{4 \cdot 151}

\sqrt{4} \cdot \sqrt{151}

2 \cdot \sqrt{151}

ⅰ

2 \cdot 15

2 \cdot (10 + 5)

2 \cdot 10 + 2 \cdot 5

20 + 10

30

x_{2} = \frac{- 2 + 2 \cdot \sqrt{151}}{30}

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

\frac{- 2 + 2 \cdot \sqrt{151}}{30}

\frac{- 1 + \sqrt{151}}{15}

x_{2} = \frac{- 1 + \sqrt{151}}{15}

▶Solutions obtained:

x = {- \frac{1 + \sqrt{151}}{15}, \frac{\sqrt{151} - 1}{15}}

2) The numeric value of:

x = {- \frac{1 + \sqrt{151}}{15}, \frac{\sqrt{151} - 1}{15}}

equals:

x = {\approx - \frac{1 + 12.2882}{15}, \approx \frac{12.2882 - 1}{15}}

x = {\approx - \frac{13.2882}{15}, \approx \frac{11.2882}{15}}

▷

x = {\approx -0.8859, \approx 0.7525}