factor 5x^3+30x^2-10x — step-by-step solution

1) Let's factor in

ℝ

5 x^{3} + 30 x^{2} - 10 x

You can write the expression in factored form:

x \cdot (5 x^{2} + 30 x - 10)

Let's factor in

ℝ

(5 x^{2} + 30 x - 10)

We factor out

5

, the greatest common divisor of the coefficients:

5 x^{2} + 30 x - 10

5 \cdot (x^{2} + 6 x - 2)

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 6

and

c = -2

Δ = 6^{2} - 4 \cdot 1 \cdot (-2)

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

6^{2}

6 \cdot 6

36

Δ = 36 - 4 \cdot (-2)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-2)

8

Δ = 36 + 8

ⓘ Start from

36

and move forward

8

on the number line:

36 + 8

44

Δ = 44

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 6 - \sqrt{44}}{2 \cdot 1}

x_{1} = \frac{- 6 - \sqrt{44}}{2}

ⓘ We look for the largest square dividing

44

4

2^{2}

, and take it out of the root:

\sqrt{44}

\sqrt{4 \cdot 11}

\sqrt{4} \cdot \sqrt{11}

2 \cdot \sqrt{11}

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

- 6 - 2 \cdot \sqrt{11}

- (6 + 2 \cdot \sqrt{11})

x_{1} = - \frac{6 + 2 \cdot \sqrt{11}}{2}

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

- \frac{6 + 2 \cdot \sqrt{11}}{2}

- \frac{2 \cdot (3 + \sqrt{11})}{2}

- (3 + \sqrt{11})

x_{1} = - (3 + \sqrt{11})

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (3 + \sqrt{11})

- 3 - \sqrt{11}

x_{1} = - 3 - \sqrt{11}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 6 + \sqrt{44}}{2 \cdot 1}

x_{2} = \frac{- 6 + \sqrt{44}}{2}

ⓘ We look for the largest square dividing

44

4

2^{2}

, and take it out of the root:

\sqrt{44}

\sqrt{4 \cdot 11}

\sqrt{4} \cdot \sqrt{11}

2 \cdot \sqrt{11}

x_{2} = \frac{- 6 + 2 \cdot \sqrt{11}}{2}

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

\frac{- 6 + 2 \cdot \sqrt{11}}{2}

\frac{2 \cdot (- 3 + \sqrt{11})}{2}

- 3 + \sqrt{11}

x_{2} = - 3 + \sqrt{11}

Factors identified:

(x - (\sqrt{11} - 3)) \cdot (x - (- \sqrt{11} - 3))

▷The factor

(5 x^{2} + 30 x - 10)

factors as:

5 \cdot (x - (\sqrt{11} - 3)) \cdot (x - (- \sqrt{11} - 3))

2) Let's replace each factor by its factorization:

x \cdot 5 \cdot (x - (\sqrt{11} - 3)) \cdot (x - (- \sqrt{11} - 3))

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (\sqrt{11} - 3)

- \sqrt{11} + 3

- (- \sqrt{11} - 3)

\sqrt{11} + 3

x \cdot 5 \cdot (x - \sqrt{11} + 3) \cdot (x + \sqrt{11} + 3)

ⓘ Factor out the common factor

x

▶Factored expression:

5 x \cdot (x + \sqrt{11} + 3) \cdot (x - \sqrt{11} + 3)