I need to learn how to write quadratic equations with given roots. It would be REALLY helpful if someone could show me a STEP BY STEP procedure in writing it in a quadratic equation.
Here are some ones I'm struggling with:
1.) -2/3 & -4/5
2.) -2+7i & -2-7i
3.) 3-sq root of 6 & 3+sq root of 6
If you could help me it would be so much appreciated! Thank you!
Here are some ones I'm struggling with:
1.) -2/3 & -4/5
2.) -2+7i & -2-7i
3.) 3-sq root of 6 & 3+sq root of 6
If you could help me it would be so much appreciated! Thank you!
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First, add the opposite of those 2 factors. So if the factors are (-2/3) and (-4/5), the equation is (x + 2/3)(x + 4/5)
The distribute and you get x^2 + (22/15)x + 8/15
Then multiply everything by 15.
15x^2 + 22x + 8
For the second one, do the same thing
(x -(-2 + 7i))(x -(-2 - 7i))
(x + 2 - 7i)(x + 2 + 7i)
x^2 + 2x + 7ix + 2x + 4 + 14i - 7ix - 14i - 49i^2
Combine like terms and you get:
x^2 + 4x + 49
For the third one, do the same thing (v means square root)
(x - (3 - v6))(x -(3 + v6)
(x - 3 + v6)(x - 3 - v6)
x^2 - 3x - xv6 - 3x + 9 + 3v6 + xv6 - 3v6 - 6
x^2 - 6x + 3
I hope this information was very helpful.
The distribute and you get x^2 + (22/15)x + 8/15
Then multiply everything by 15.
15x^2 + 22x + 8
For the second one, do the same thing
(x -(-2 + 7i))(x -(-2 - 7i))
(x + 2 - 7i)(x + 2 + 7i)
x^2 + 2x + 7ix + 2x + 4 + 14i - 7ix - 14i - 49i^2
Combine like terms and you get:
x^2 + 4x + 49
For the third one, do the same thing (v means square root)
(x - (3 - v6))(x -(3 + v6)
(x - 3 + v6)(x - 3 - v6)
x^2 - 3x - xv6 - 3x + 9 + 3v6 + xv6 - 3v6 - 6
x^2 - 6x + 3
I hope this information was very helpful.
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x = -2/3
x * 3 = -2/3 * 3
3x = -2
3x + 2 = -2 + 2
3x+2 = 0
One of the factors is 3x+2.
x = -4/5
x * 5 = -4/5 * 5
5x = -4
5x + 4 = -4 + 4
5x+4 = 0
Another factor is 5x+4.
(3x+2)(5x+4) = 0
15x^2+12x+10x+8 = 0
15x^2+22x+8 = 0
Answer: 15x^2+22x+8=0
______________________________________…
x = -2+7i
x = -2+7√-1
x = -2+√49√-1
x = -2+√-49
x + 2 = -2+√-49 + 2
x+2 = √-49
(x+2)^2 = √-49^2
(x+2)(x+2) = -49
x^2+2x+2x+4 = -49
x^2+4x+4 = -49
x^2+4x+4 + 49 = -49 + 49
x^2+4x+53 = 0
One of the factors is x^2+4x+53. The other factor is also the same but I'll show you why. So really, instead of being a factor, this is the actual function.
x = -2-7i
x = -2-7√-1
x = -2-√49√-1
x = -2-√-49
x + 2 = -2-√-49 + 2
x+2 = -√-49
(x+2)^2 = (-√-49)^2
(x+2)^2 = -49
(x+2)(x+2) = -49
x^2+2x+2x+4 = -49
x^2+4x+4 = -49
x^2+4x+4 + 49 = -49 + 49
x^2+4x+53 = 0
See? It's the same.
Answer: x^2+4x+53=0
______________________________________…
x = 3-√6
x - 3 = 3-√6 - 3
x-3 = -√6
(x-3)^2 = (-√6)^2
(x-3)(x-3) = 6
x^2-3x-3x+9 = 6
x^2-6x+9 = 6
x^2-6x+9 - 6 = 6 - 6
x^2-6x+3 = 0
This is like the previous problem. But I'll show you anyway.
x = 3+√6
x -3 = 3+√6 - 3
x-3 = √6
(x-3)^2 = √6^2
(x-3)(x-3) = 6
x^2-3x-3x+9 = 6
x^2-6x+9 = 6
x^2-6x+9 - 6 = 6 - 6
x^2-6x+3 = 0
See? You can even use the quadratic formula to check.
Answer: x^2-6x+3=0
x * 3 = -2/3 * 3
3x = -2
3x + 2 = -2 + 2
3x+2 = 0
One of the factors is 3x+2.
x = -4/5
x * 5 = -4/5 * 5
5x = -4
5x + 4 = -4 + 4
5x+4 = 0
Another factor is 5x+4.
(3x+2)(5x+4) = 0
15x^2+12x+10x+8 = 0
15x^2+22x+8 = 0
Answer: 15x^2+22x+8=0
______________________________________…
x = -2+7i
x = -2+7√-1
x = -2+√49√-1
x = -2+√-49
x + 2 = -2+√-49 + 2
x+2 = √-49
(x+2)^2 = √-49^2
(x+2)(x+2) = -49
x^2+2x+2x+4 = -49
x^2+4x+4 = -49
x^2+4x+4 + 49 = -49 + 49
x^2+4x+53 = 0
One of the factors is x^2+4x+53. The other factor is also the same but I'll show you why. So really, instead of being a factor, this is the actual function.
x = -2-7i
x = -2-7√-1
x = -2-√49√-1
x = -2-√-49
x + 2 = -2-√-49 + 2
x+2 = -√-49
(x+2)^2 = (-√-49)^2
(x+2)^2 = -49
(x+2)(x+2) = -49
x^2+2x+2x+4 = -49
x^2+4x+4 = -49
x^2+4x+4 + 49 = -49 + 49
x^2+4x+53 = 0
See? It's the same.
Answer: x^2+4x+53=0
______________________________________…
x = 3-√6
x - 3 = 3-√6 - 3
x-3 = -√6
(x-3)^2 = (-√6)^2
(x-3)(x-3) = 6
x^2-3x-3x+9 = 6
x^2-6x+9 = 6
x^2-6x+9 - 6 = 6 - 6
x^2-6x+3 = 0
This is like the previous problem. But I'll show you anyway.
x = 3+√6
x -3 = 3+√6 - 3
x-3 = √6
(x-3)^2 = √6^2
(x-3)(x-3) = 6
x^2-3x-3x+9 = 6
x^2-6x+9 = 6
x^2-6x+9 - 6 = 6 - 6
x^2-6x+3 = 0
See? You can even use the quadratic formula to check.
Answer: x^2-6x+3=0