02.07 The Discriminant

VIDEO LESSON

INTERACTIVE SELF-STUDY

We want to find a quick way to determine whether a quadratic equation has any solutions or roots and, if so, how many. We can derive a condition geometrically and algebraically.

INVESTIGATION

Let’s begin with the geometric derivation.

Let \(f(x)=ax^2+bx+c=0\).
Recall that we can complete the square to obtain \(\displaystyle f(x)=a\left(x+\frac{b}{2a}\right)^2+\left(c−\frac{b^2}{4a}\right)\)
i. What are the coordinates of the turning point of this quadratic? Show Answer

Suppose that \(a>0\) so that we have a positive quadratic and hence a minimum point. The position of this minimum point tells us how many solutions there are.

What can you say about the \(y\)-coordinate of the minimum point when there are:
ii. Two distinct real roots to the quadratic equation \(f(x)=ax^2+bx+c=0\) Show Answer

v. Hence, using the \(y\)-coordinate of the turning point, \(\displaystyle c−\frac{b^2}{4a}\), derive an equation or inequality for each case above and rearrange it so that it is in terms of \(b^2−4ac\) Show Answer

We can also see the significance of \(b^2−4ac\) algebraically from the quadratic formula.

The two roots given by the quadratic formula are \(\displaystyle α= -\frac{b}{2a}-\frac{\sqrt{b^2-4ac}}{2a}\) and \(\displaystyle β=-\frac{b}{2a}+\frac{\sqrt{b^2-4ac}}{2a}\)

Both roots have \(\displaystyle -\frac{b}{2a}\) in common, but are distinguished by the addition and subtraction of \(\displaystyle \frac{\sqrt{b^2-4ac}}{2a}\)

1. We have to be careful when square rooting. We cannot square root negative numbers so if the value under the square root, \(b^2−4ac\) , is negative, then we have no real solutions
2. If the value under the square root, \(b^2−4ac\), is \(0\), then we will just be adding \(0\) or subtracting \(0\) from \(\displaystyle -\frac{b}{2a}\) hence the roots \(α\) and \(β\) will be identical and we just have one repeated real root or equal roots
3. If the value under the square root, \(b^2−4ac\), is positive, then we will be adding/subtracting something positive to/from \(\displaystyle -\frac{b}{2a}\) hence the roots \(α\) and \(β\) will be distinct and we have two distinct real roots

• Hence \(b^2−4ac\) is called the discriminant. It discriminates or distinguishes the number of solutions to a quadratic equation.
  It is sometimes represented by the symbol \(\Delta\)

• If \(b^2−4ac>0\), then the quadratic equation \(f(x)=ax^2+bx+c=0\) has two distinct real roots
• If \(b^2−4ac=0\), then the quadratic equation \(f(x)=ax^2+bx+c=0\) has one repeated real root (equal roots)
• If \(b^2−4ac<0\), then the quadratic equation \(f(x)=ax^2+bx+c=0\) has no real roots

Example 1 – (GRADE D)
Consider the function \(f(x)=3x^2-4x-9\)
a. Calculate the discriminant of the function Show Answer

Example 2 – (GRADE D)
a. Find the value(s) of \(k\) such that \(2x^2+kx+18=0\) has exactly one solution Show Answer

END OF LESSON!

You can now answer:
EDEXCEL Pure Year 1 Ex 2F – All questions
OCR A Student Book 1 Ex 3E – Q3c Q4 Q5

INVESTIGATION

Let’s do a geometric derivation of the discriminant for the case \(a<0\) and show that it leads to the same result!

Suppose that \(a<0\) so that our quadratic function has a maximum point. Recall that the coordinates of the turning point is \(\displaystyle \left(-\frac{b}{2a}, c−\frac{b^2}{4a}\right)\)

State the conditions on the coordinate(s) of the maximum point so that the quadratic equation \(f(x)=ax^2+bx+c=0\) has:
i. Two distinct real roots Show Answer