2x^2+13x-15=0 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} + 13 x - 15 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 13

and

c = -15

Δ = 13^{2} - 4 \cdot 2 \cdot (-15)

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

13^{2}

13 \cdot 13

169

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

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

- 8 \cdot (-15)

8 \cdot 15

Δ = 169 + 8 \cdot 15

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 15

8 \cdot (10 + 5)

8 \cdot 10 + 8 \cdot 5

80 + 40

120

Δ = 169 + 120

ⓘ Start from

169

and move forward

120

on the number line:

169 + 120

289

Δ = 289

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 13 - \sqrt{289}}{2 \cdot 2}

x_{1} = \frac{- 13 - \sqrt{289}}{2 \cdot 2}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 2

4

x_{1} = \frac{- 13 - 17}{4}

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

- 13 - 17

-30

x_{1} = - \frac{30}{4}

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

2

, their greatest common divisor:

- \frac{30}{4}

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

- \frac{15}{2}

x_{1} = - \frac{15}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 13 + \sqrt{289}}{2 \cdot 2}

x_{2} = \frac{- 13 + \sqrt{289}}{2 \cdot 2}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 2

4

x_{2} = \frac{- 13 + 17}{4}

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

- 13 + 17

4

x_{2} = \frac{4}{4}

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

4

, their greatest common divisor:

\frac{4}{4}

1

x_{2} = 1

▶Solutions found:

x = {- \frac{15}{2}, 1}

2) The numeric value of:

x = {- \frac{15}{2}, 1}

is:

▷

x = {-7.5, 1}