Factor the quadratic polynomial completely:
4x² + 16x – 8 = 0 (i)
4x² + 4 · 4x + 4 · (– 2) = 0 (i)
Take out the common fator 4 at the left-hand side:
4 · (x² + 4x – 2) = 0 (ii)
Finding the roots using the quadratic formula:
x² + 4x – 2 = 0 ———> a = 1, b = 4, c = – 2
Δ = b² – 4ac
Δ = 4² – 4 · 1 · (– 2)
Δ = 16 + 8
Δ = 24
Δ = 2³ · 3
Δ = 2² · 2 · 3
Δ = 2² · 6
Therefore, the roots are
– b – √Δ – b + √Δ
r₁ = —————— and r₂ = ——————
2a 2a
– 4 – √(2² · 6) – 4 + √(2² · 6)
r₁ = ———————— and r₂ = ————————
2 · 1 2 · 1
– 4 – 2√6 – 4 + 2√6
r₁ = —————— and r₂ = ——————
2 2
Take out the common factor 2 in both numerators, and then simplify those fractions:
2 · (– 2 – √6) 2 · (– 2 + √6)
r₁ = ———————— and r₂ = ————————
2 2
r₁ = – 2 – √6 and r₂ = – 2 + √6 <——— roots of equation (ii)
So, the factored form of the left-hand side in (ii) is
4 · (x – r₁) · (x + r₁)
= 4 · [ x – (– 2 – √6) ] · [ x – (– 2 + √6) ]
= 4 · (x + 2 + √6) · (x + 2 – √6) <——— complete factored form
(this is the anwer)
I hope this helps. =)
