# find the equation of a quadratic with roots (1+root 5) and (1-root5) which passes through the point (2,5)?

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• The roots are 1+√5 and 1-√5

f(x) = a(x-(1+√5))(x-(1-√5)) = a((x-1)-√5)((x-1)+√5) =a((x-1)^2-5) = a(x^2-2x+1-5) = a(x^2-2x-4)

5 = a(2^2-2(2)-4) = -4a

a= -5/4

f(x) = -5/4(x^2-2x-4)= (-5/4)x^2+(5/2)x+5

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• y = 5/(((1+√5) - 2)((1-√5) - 2)) ((1+√5) - x)((1-√5) - x)

y = 5/(-4) (x² - 2x - 4)

y = -⁵∕₄ x² + ⁵∕₂ x + 5

Source(s): The equation of the polynomial curve with off x-axis point (a, b), and x intercepts x₀, x₁, etc. (not equal to a ) is y = (b/∏(xᵢ-a)) ∏(xᵢ-x).
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• .

Quadratic equation has a form of;

y = ax² + bx² + c

Using roots m and n, the quadratic equation can be derived from the following:

y = A( x - m )( x - n )

Distinct roots:

x = 1 + √5 =====> ( x - 1 - √5 )

x = 1 - √5 =====> ( x - 1 + √5 )

Resulting equation

y = A( x - 1 - √5 )( x - 1 + √5 )

y = A( x² - x + x√5 - x + 1 - √5 - x√5 + √5 - 5 )

y = A( x² - 2x - 4 )

but passes through (2, 5)

y = A( x² - 2x - 4 )

5 = A( 2² - 2*2 - 4 )

A = -5/4

y = -5/4( x² - 2x - 4 )

━━━━━━━━━

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• f(x) = a((x-1)+√5)((x-1)-√5)

= a(x-1)² - 5)

= a(x² - 2x - 4)

f(2) = 5

5 = a(4 - 4 - 4)

a = -5/4

f(x) = -5x²/4 + 5x/2 + 5

or

f(x) = -5(x/2)² + 5(x/2) + 5

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• roots (1 + √5) and (1 - √5)

that means the equation is

(x – (1 + √5))(x – (1 - √5)) = 0

(x – 1 – √5)(x – 1 + √5) = 0

x² – x + x√5 – x + 1 – √5 –x√5 + √5 – 5 = 0

x² – 2x – 4 = 0

ky = x² – 2x – 4

plug in point

k•5 = 2² – 2•2 – 4

k•5 = 4 – 4 – 4

k = –5/4

so equation is

–(5/4)y = x² – 2x – 4

or

–5y = 4x² – 8x – 16

• Puzzling
Lv 7
6 months agoReport

Double-check your math at the end. If you plug in x=2, you don't get y = 5.

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• With those roots:

y = a(x - (1 + sqrt(5)))(x - (1 - sqrt(5)))

y = a(x - 1 - sqrt(5))(x - 1 + sqrt(5))

y = a(x^2 - x + sqrt(5)*x - x + 1 - sqrt(5) - sqrt(5)*x + sqrt(5) - 5)

y = a(x^2 - 2x - 4)

Passing through that point:

5 = a(2^2 - 2*2 - 4)

5 = a(4 - 4 - 4)

-4a = 5

a = -5/4 = -1.25

y = -1.25x^2 + 2.5x + 5

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• f(x) = (x - (1 + √5))(x - (1 - √5))

f(x) = (x - 1 - √5))(x - 1 + √5)

f(x) = ((x - 1) - √5)) ((x - 1) + √5)

f(x) = (x - 1)^2 - √5^2

f(x) = x^2 - 2x - 4

• mizoo
Lv 7
6 months agoReport

I had missed the last sentence of your question. Above, I added some steps and final answer is (-5/4)(x^2 - 2x - 4). Hope this helps.

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• y = A ( ( x-1)² - 5 ) where A = - 5 / 4

• Sally6 months agoReport

how did u get there

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