Week 7 – Interpreting the Discriminant

Posted on by shannonf2015

Last week, we have learned how to determine the number of solutions of a quadratic equation without solving the equation. To be able to do that, we look at the discriminant!

The discriminant is the radicand of the quadratic formula: b^2-4ac

This is how to determine the number of solutions by the discriminant.

  • Two real roots when b^2-4ac > 0

If the discriminant is positive (so greater than 0), it has two solutions. (It touches the x-axis twice)

  • Exactly one real root when b^2-4ac = 0

If the discriminant is zero, that means there is exactly one solution (Just touches the x-axis once)

  • No real roots when b^2-4ac < 0

If the discriminant is a negative number (so smaller than 0), it has no solutions (Does not touch the x-axis)

Now that we understand how to use the discriminant to determine the number of solutions, let’s apply these to some equations.

First example: Calculate the value of the discriminant and determine how many solutions.

4x^2+2x+3=0

Remember the quadratic equation is : ax^2+bx+c=0

Therefore: a = 4, b = 2, c = 3

Plug in the numbers to determine the discriminant.

b^2-4ac

2^2-4(4)(3)

4-16(3)

4-48

-44

Since the discriminant is negative, it has no solutions; no real roots.

Now let’s look at another example where we have to create an equation with a given number of roots.

Determine the values of y of the equation which has two real roots.

5x^2+3x+y=0

To have two real roots, the value of its discriminant must be greater than 0.

So we use: b^2-4ac > 0

Substitute: a = 5, b = 3, and c = y

3^2-4(5)(y) > 0

Simplify.

9 – 20y > 0

Move the 9 to the other side to isolate the variable.

-20y > -9

Divide both sides by -20.

y > \frac{9}{20}

Since we divided both sides by a negative number, we must switch the inequality symbol.

y < \frac{9}{20}