# Solving Quadratic Equations w/ Unequal, Real, Rational Roots

• mrroboto
In summary, the discriminant for this problem is a perfect square. If the roots are real and rational, the discriminant can't be negative, and if the roots are unequal, the discriminant can't be a perfect square.
mrroboto

## Homework Statement

Barry has just solved a quadratic equation. He sees that the roots are rational, real, and unequal. This means the discriminant is

a) zero, b) negative, c) a perfect square, d) a non perfect square

## The Attempt at a Solution

I think the answer is d) a non perfect square

if the roots are real and rational then the discriminant can't be negative, and if they are unequal then the discriminant can't be a perfect square

is this the right way to do this problem?

mrroboto said:
1.

I think the answer is d) a non perfect square

if the roots are real and rational then the discriminant can't be negative, and if they are unequal then the discriminant can't be a perfect square

is this the right way to do this problem?

Why do you think so?? You are expected to show your work before anyone here can help you! Ok, the general form of the quadratic eq is:

$$ax^{2}+bx+c=0$$ the formula for the discriminant is

$$D=b^{2}-4ac$$ right?

The formula for the two roots is:

$$x_1,_2=\frac{-b+-\sqrt D}{2a}$$, so you want your answer to be a rational nr, and the roots to be distinct, right?
This means:

$$x_1=\frac{-b-\sqrt D}{2a}= \frac{m}{n} \ (not \ equal \ to)=/=x_2=\frac{-b+\sqrt D}{2a}=\frac{p}{q}$$ where m,n,p,q are integers.

So what do you think now? What would happen if, say D=3, D=4, D=0, or D<0??

Last edited:
Not exactly. Look at the general solution to a quadratic equation of one variable. What kind of solution occurs when the discriminant fits each of the choices in your question? What kind of discriminant will give you TWO solutions which are rational and real and unequal?

I see. the discriminant should be a perfect square. thanks.

mrroboto said:
I see. the discriminant should be a perfect square. thanks.

You're welcome! Just make sure to show some work of yours next time!

## 1. What is a quadratic equation?

A quadratic equation is a polynomial 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 has a degree of 2.

## 2. How do you solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2-4ac)) / 2a. You can also solve it by factoring the equation and setting each factor equal to 0. Another method is completing the square, where you add and subtract a value to make the equation a perfect square.

## 3. What does it mean when a quadratic equation has unequal roots?

When a quadratic equation has unequal roots, it means that the equation has two distinct solutions. This occurs when the discriminant, b^2-4ac, is greater than 0.

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

A quadratic equation has real roots if the discriminant, b^2-4ac, is greater than or equal to 0. If the discriminant is a perfect square, the roots will be rational. If the discriminant is not a perfect square, the roots will be irrational.

## 5. Can a quadratic equation have rational roots?

Yes, a quadratic equation can have rational roots. This occurs when the discriminant, b^2-4ac, is a perfect square. In this case, the solutions will be rational numbers. However, a quadratic equation can also have irrational roots if the discriminant is not a perfect square.

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