Finding the roots of a quadratic equation

In summary, the conversation involves finding the working for a quadratic equation and checking its correctness. The person initially asks for their working to be checked and then proceeds to demonstrate the correct form for the equation. They also mention using the same thinking for different values of α and β. However, there is a question about the possibility of letting α=β.
  • #1
chwala
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Homework Statement
Kindly see the attached problem below
Relevant Equations
sum and products of roots of a quadratic equation
1617241655897.png
 
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  • #2
find my working on this below;

1617241722760.png

1617241762395.png

i would like you to check my working...is it correct?
 
  • #3
After you find k (which you do correctly) just solve the equation explicitly as a check. You don't need my help.
 
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  • #4
i should be correct, checking with a previous example...
1617246725763.png


if i let ##α=3## and ##β=1##, then ##(x-3)(x-1)=x^2-4x+3##, ##p=4## and ##q=3##
on forming the required quadratic with form ##\frac {α}{β^2}## and ##\frac {β}{α^2}##,
we shall have,
##(x-\frac {1}{9}##)##(x-3)##=##x^2-\frac {28}{9}x##+##\frac {1}{3}## which is in the required form...
similarly using the same thinking, and picking the roots ##α=-3## and ##β=-3## would yield the required form that i had shown in my working...
problem is can we let ##α=β?##
 
Last edited:
  • #5
chwala said:
i should be correct, checking with a previous example...
View attachment 280707

if i let ##∝=3## and ##β=1##,
Where does that combination come from? Did you mean α=β=3?
Btw, ∝ means "is proportional to". It is not a form of α (or in LaTeX, ##\alpha##).
 
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  • #6
haruspex said:
Where does that combination come from? Did you mean α=β=3?
Btw, ∝ means "is proportional to". It is not a form of α (or in LaTeX, α).

yeah, let me amend that...my eyes did not see that well..am getting oldo0)
 

1. What is a quadratic equation?

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It represents a parabola when graphed and can have two solutions, or roots, when solved.

2. How do you find the roots of a quadratic equation?

The roots of a quadratic equation can be found by using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula can be used when the equation is in standard form, or by factoring the equation and setting each factor equal to 0.

3. Can a quadratic equation have more than two roots?

No, a quadratic equation can only have a maximum of two roots. This is because a quadratic equation is a second-degree polynomial and the Fundamental Theorem of Algebra states that a polynomial of degree n can have at most n roots.

4. How do you know if a quadratic equation has real or imaginary roots?

A quadratic equation has real roots if the discriminant, b^2 - 4ac, is greater than or equal to 0. If the discriminant is less than 0, the equation will have imaginary roots. The square root of a negative number is not a real number, so the roots will be in the form of a complex number.

5. Can the roots of a quadratic equation be irrational numbers?

Yes, the roots of a quadratic equation can be irrational numbers. This means that the roots cannot be expressed as a fraction or decimal and will have an infinite number of non-repeating digits after the decimal point. For example, the roots of x^2 - 2 = 0 are ±√2, which are irrational numbers.

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