4x^2+3x-10=0 — step-by-step solution

1) Let's solve in

ℝ

4 x^{2} + 3 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 = 4

b = 3

and

c = -10

Δ = 3^{2} - 4 \cdot 4 \cdot (-10)

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

3^{2}

3 \cdot 3

9

■Multiplications are done before additions and subtractions:

4 \cdot 4

16

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

- 16 \cdot (-10)

16 \cdot 10

Δ = 9 + 16 \cdot 10

ⓘ Multiplications are done before additions and subtractions:

16 \cdot 10

(10 + 6) \cdot 10

10 \cdot 10 + 6 \cdot 10

100 + 60

160

Δ = 9 + 160

ⓘ Start from

9

and move forward

160

on the number line:

9 + 160

169

Δ = 169

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 3 - \sqrt{169}}{2 \cdot 4}

x_{1} = \frac{- 3 - \sqrt{169}}{2 \cdot 4}

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

\sqrt{169}

13

;

13^{2}

169

ⅰ

2 \cdot 4

8

x_{1} = \frac{- 3 - 13}{8}

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

- 3 - 13

-16

x_{1} = - \frac{16}{8}

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

8

, their greatest common divisor:

- \frac{16}{8}

-2

x_{1} = -2

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 3 + \sqrt{169}}{2 \cdot 4}

x_{2} = \frac{- 3 + \sqrt{169}}{2 \cdot 4}

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

\sqrt{169}

13

;

13^{2}

169

ⅰ

2 \cdot 4

8

x_{2} = \frac{- 3 + 13}{8}

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

- 3 + 13

10

x_{2} = \frac{10}{8}

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

2

, their greatest common divisor:

\frac{10}{8}

\frac{5 \cdot 2}{4 \cdot 2}

\frac{5}{4}

x_{2} = \frac{5}{4}

▶Solutions obtained:

x = {-2, \frac{5}{4}}

2) The numeric value of:

x = {-2, \frac{5}{4}}

is:

▷

x = {-2, 1.25}