Question

Zapisz w postaci iloczynu.

1. x^2 - 2x -3

2. y^2-4y-5

3.z^2-2x^2

4.3-w^2-2w

Answer 1

Δ=b²-4ac

x₁=[-b-√Δ]/2a

x₂=[-b+√Δ]/2a

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1. x^2 - 2x -3=(x+1)(x-3)

Δ=b²-4ac=(-2)²-4*1*(-2)=4+12=16

√Δ=4

x₁=[-b-√Δ]/2a=[2-4]/2=-1

x₂=[-b+√Δ]/2a=[2+4]/2=3

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2. y^2-4y-5=(y+1)(y-5)

Δ=b²-4ac=(-4)²-4*1*(-5)=16+20=36

√Δ=6

y₁=[-b-√Δ]/2a=[4-6]/2=-1

y₂=[-b+√Δ]/2a=[4+6]/2=5

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3.z^2-2x^2=(z-x√2)(z+x√2)

Wzór: a²-b²=(a-b)(a+b)

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4.3-w^2-2w=-w²-2w+3=(w-1)(w+3)

Δ=b²-4ac=(-2)²-4*(-1)*3=4+12=16

√Δ=4

w₁=[-b-√Δ]/2a=[2-4]/-2=1

w₂=[-b+√Δ]/2a=[2+4]/-2=-3

Answer 2

x² - 2x - 3

Δ = 4 - 4 * 1 * (-3) = 4 + 12 = 16

√Δ = 4

x1 = (2 - 4)/2 = -2/2 = -1

x2 = (2 + 4)/2 = 6/2 = 3

 odp.  x² - 2x - 3 = (x + 1)(x - 3) 

 

 

y² - 4y - 5

Δ = 16 - 4 * 1 * (-5) = 16 + 20 = 36

√Δ = 6

y1 = (4 - 6)/2 = -2/2 = -1

y2 = (4 + 6)/2 = 10/2 = 5

odp.  y² - 4y - 5 = (y + 1)(y - 5)

 

z² - 2x² = z² - (√2x)² = (z - √2x)(z + √2x) 

 

 

3 - w² - 2w =  -w² - 2w + 3 = w² + 2w - 3 

Δ = 4 - 4 * 1 * (-3) = 4 + 12 = 16

√Δ = 4

w1 =  (-2 - 4)/2 = -6/2 = -3

w2 = (-2 + 4)/2 = 2/2 = 1

odp. w² + 2w - 3 = (w + 3)(w - 1)