Week 7– precalculus 11

Discriminant:

This week in precalculus we learned about the discriminant of the quadratic formula and how to use it to find information. By using the formula $b^2 - 4ac$, we can find the discriminant. This section of the quadratic formula tells us whether a quadratic equation has zero, one, or two roots. We call this the discriminant of the quadratic equation because it is discriminating the different varieties of solutions. The discriminant tells us different things based off of what it equals.

When $b^2 - 4ac = 0$ There is only 1 real root,

When $b^2 - 4ac > 0$ There is two roots, if it is a perfect square, it is factorable, If not then it is not factorable

and When When $b^2 - 4ac < 0$ There is no real roots. A radicand cannot be negative.

This is because the discriminant equation is being square rooted.

If we are given the equation $3^2 - 4x + k = 0$ and are told it must have two roots, we can find what k will be by finding the discriminant. $(-4)^2 - 4 (3)(k) > 0$

Then $16 - 12k > 0$

Then we put 12k on the right side

16 > 12k

Then $\frac {4}{3} > k$

So k is less than $\frac {4}{3}$ if the equation is to have two real roots. A possible value is 1 for k.

To see if a quadratic is factorable or not we will use the equation $x^2 - 11x + 30 = 0$, we can plug this into the Discriminant equation, $b^2 - 4ac$ $(-11)^2 - 4(1)(30)$

Then we get $121 -120 = 1$

In this case, there will be 2 roots but because 1 is a perfect square, we know this is factorable and can use our normal factoring method to find the values of x. If it was not a perfect square we would have to use the completing the square or Quadratic Formula methods to find x