factor 5x^2+2x-3 — step-by-step solution

Let's factor in

ℝ

5 x^{2} + 2 x - 3

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 5

b = 2

and

c = -3

Δ = 2^{2} - 4 \cdot 5 \cdot (-3)

ⓘ 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 5

20

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

- 20 \cdot (-3)

20 \cdot 3

Δ = 4 + 20 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

20 \cdot 3

60

Δ = 4 + 60

ⓘ Start from

4

and move forward

60

on the number line:

4 + 60

64

Δ = 64

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 2 - \sqrt{64}}{2 \cdot 5}

x_{1} = \frac{- 2 - \sqrt{64}}{2 \cdot 5}

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

\sqrt{64}

8

;

8^{2}

64

ⅰ

2 \cdot 5

10

x_{1} = \frac{- 2 - 8}{10}

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

- 2 - 8

-10

x_{1} = - \frac{10}{10}

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

10

, their greatest common divisor:

- \frac{10}{10}

-1

x_{1} = -1

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 2 + \sqrt{64}}{2 \cdot 5}

x_{2} = \frac{- 2 + \sqrt{64}}{2 \cdot 5}

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

\sqrt{64}

8

;

8^{2}

64

ⅰ

2 \cdot 5

10

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

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

- 2 + 8

6

x_{2} = \frac{6}{10}

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

2

, their greatest common divisor:

\frac{6}{10}

\frac{3 \cdot 2}{5 \cdot 2}

\frac{3}{5}

x_{2} = \frac{3}{5}

A trinomial with roots

x_{1}

and

x_{2}

a \cdot (x - x_{1}) \cdot (x - x_{2})

; bringing the coefficient into the brackets:

5 \cdot (x + 1) \cdot (x - \frac{3}{5})

(x + 1) \cdot (5 x - 3)

▶Factors found:

(x + 1) \cdot (5 x - 3)