Question

lesson 3-8: writing quadratics from root (complex)
practice problems
algebra 2

write the equation of the quadratic from the given solutions:
1.) ( x = 2 pm sqrt{3} )
2.) ( x = 6 pm sqrt{10} )

3.) ( x = 3 pm 2i )
4.) ( x = 1 pm i )

5.) ( x = -1 pm sqrt{1} )
6.) ( x = 1 pm sqrt{-1} )

Explanation:

For each problem, we use the fact that if $x = h \pm k$, then $(x - h)^2 = k^2$, which rearranges to $x^2 - 2hx + (h^2 - k^2) = 0$.

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Problem 1: $x = 2 \pm \sqrt{3}$

Step1: Set up squared difference

$(x - 2)^2 = (\sqrt{3})^2$

Step2: Expand and simplify

$x^2 - 4x + 4 = 3$
$x^2 - 4x + 1 = 0$

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Problem 2: $x = 6 \pm \sqrt{10}$

Step1: Set up squared difference

$(x - 6)^2 = (\sqrt{10})^2$

Step2: Expand and simplify

$x^2 - 12x + 36 = 10$
$x^2 - 12x + 26 = 0$

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Problem 3: $x = 3 \pm 2i$

Step1: Set up squared difference

$(x - 3)^2 = (2i)^2$

Step2: Expand and simplify

$x^2 - 6x + 9 = -4$
$x^2 - 6x + 13 = 0$

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Problem 4: $x = 1 \pm i$

Step1: Set up squared difference

$(x - 1)^2 = (i)^2$

Step2: Expand and simplify

$x^2 - 2x + 1 = -1$
$x^2 - 2x + 2 = 0$

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Problem 5: $x = -1 \pm \sqrt{1}$

Step1: Simplify the root first

$\sqrt{1}=1$, so $x = -1 \pm 1$

Step2: Set up squared difference

$(x + 1)^2 = (1)^2$

Step3: Expand and simplify

$x^2 + 2x + 1 = 1$
$x^2 + 2x = 0$

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Problem 6: $x = 1 \pm \sqrt{-1}$

Step1: Simplify the root first

$\sqrt{-1}=i$, so $x = 1 \pm i$

Step2: Set up squared difference

$(x - 1)^2 = (i)^2$

Step3: Expand and simplify

$x^2 - 2x + 1 = -1$
$x^2 - 2x + 2 = 0$

Answer:

1. $x^2 - 4x + 1 = 0$
2. $x^2 - 12x + 26 = 0$
3. $x^2 - 6x + 13 = 0$
4. $x^2 - 2x + 2 = 0$
5. $x^2 + 2x = 0$
6. $x^2 - 2x + 2 = 0$