Pre-Calculus 11 Notes

By: Gen L.

Lesson 4: Solving Quadratics - Formula

Terms

• Nature of the Roots: Number of solutions.
• Discriminant ():
• Exact answer: an answer w/o a decimal
• Approximate answer: an answer w/ a decimal

Why Quadratic Formula?

• Sometimes, there are equations that are too difficult to solve by factoring or graphing.
• When this occurs, it is best to use the formula:

• All thats needed is a quadratic equation in standard form.

Example: Comparing Factoring and Quadratic Formula

• Factoring

Example, cont.

• Quadratic Formula

Note:

1. We get the same answer whether we factor or use the formula.
2. Using the formula takes longer than factoring.
   • This is because the formula was intended for the equations where factoring by hand is near-impossible.
     It is just another tool in our toolkit to solve quadratic equations.

Example 2

• Find the exact zeros of

Example 3

• Find the x-intercepts of

Exceptions: 0 Real Roots

• Sometimes, the formula doesn't always "keep it real".
• Example: Solve
• This means that there are no real roots.

Exceptions: 1 Distinct Real Root

• Or it has a matching root.
• Example: Solve
• This means there is one real root.

Discriminant and Nature of The Roots

• The expression (under the radical sign) is the Discriminant ().
• This allows us to determine the Nature of The Roots for a quadratic equation without solving the equation.

Discriminant () Rules

• If , there are Distinct real roots.
• If , there is Distinct real root or equal real roots.
• If
• Which can be written like this:

Example: Nature of the Roots

• Determine the nature of the roots for . DO NOT SOLVE.

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