factor x^4+7x^2-144 — step-by-step solution

1) Let's factorize in

ℝ

x^{4} + 7 x^{2} - 144

⇥1.1) Let's define:

X = x^{2}

⇥You can rewrite the expression as:

X^{2} + 7 X - 144

⇥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 = 7

and

c = -144

⇥

Δ = 7^{2} - 4 \cdot 1 \cdot (-144)

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

7^{2}

7 \cdot 7

49

⇥

Δ = 49 - 4 \cdot (-144)

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

- 4 \cdot (-144)

4 \cdot 144

4 \cdot (100 + 40 + 4)

4 \cdot 100 + 4 \cdot 40 + 4 \cdot 4

400 + 160 + 16

576

⇥

Δ = 49 + 576

⇥ⓘ Start from

49

and move forward

576

on the number line:

49 + 576

625

⇥

Δ = 625

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

X_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 7 - \sqrt{625}}{2 \cdot 1}

⇥

X_{1} = \frac{- 7 - \sqrt{625}}{2}

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

\sqrt{625}

25

;

25^{2}

625

⇥

X_{1} = \frac{- 7 - 25}{2}

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

- 7 - 25

-32

⇥

X_{1} = - \frac{32}{2}

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

2

, their greatest common divisor:

- \frac{32}{2}

-16

⇥

X_{1} = -16

⇥The quadratic formula gives

X_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 7 + \sqrt{625}}{2 \cdot 1}

⇥

X_{2} = \frac{- 7 + \sqrt{625}}{2}

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

\sqrt{625}

25

;

25^{2}

625

⇥

X_{2} = \frac{- 7 + 25}{2}

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

- 7 + 25

18

⇥

X_{2} = \frac{18}{2}

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

2

, their greatest common divisor:

\frac{18}{2}

9

⇥

X_{2} = 9

⇥Factors found:

(X - 9) \cdot (X + 16)

⇥1.2) Let's factorize in

ℝ

(x^{2} - 9) \cdot (x^{2} + 16)

⇥Let's factorize in

ℝ

(x^{2} - 9)

⇥For

x^{2} - 9

, we recognize a difference of two squares:

x^{2} - 3^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x

and

b = 3

▷The factor

(x^{2} - 9)

factors as:

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

ⓘ A square being never negative,

x^{2} + 16

is never zero and cannot be factored in ℝ

▶Factored expression:

(x + 3) \cdot (x - 3) \cdot (x^{2} + 16)