# 3.6 – The Quadratic Formula

## Objectives

• Write the general form of the quadratic formula and state its use.
• State the conditions that must be met to use the quadratic formula.
• Identify equations that can be solved using the quadratic formula.
• Use the quadratic formula to find the solutions to a quadratic equation.
• Define the discriminant and state what it tells about the solutions to the equation.
• Apply the quadratic formula to find the roots of a quadratic equation.
• Use the discriminant to determine how many roots a polynomial has.

## Key Terms

• Complex Numbers – The set of all numbers of the form a + bi, where a and b are any real numbers and i equals $\sqrt{-1}$.
• Discriminant – The number under the radical sign in the quadratic formula. It is given by the expression $b^2-4ac$.
• Imaginary Number – The square root of a negative number. The imaginary number i is defined as the square root of –1.
• You can use $i=\sqrt{-1}$ to find the square root of any negative number.
• The solutions to the quadratic equation $ax^2+bx+c=0$ are given by the formula $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
• Radical – A term that has the form $\sqrt[n]{b}$ where b can be a number, a variable, or an expression that contains both.
• The symbol $\sqrt[n]{}$ indicates the nth root.
• The symbol $\sqrt{}$ (no n) indicates a square root.
• Roots – Values for which a function equals zero.
• The roots are also called zeros of the function.
• Any x-value at which the graph of a function crosses the x-axis is a root of the function.

## Notes

• These are the steps to derive the quadratic formula.
• If you can derive this without your notes on December 1st, you get an excused grade on a TST (meaning that your lowest TST grade is dropped)!
• You will get ONE chance, and you cannot miss a step!

• To solve quadratic equations, you can:
• factor and use the zero product rule (Unit 3.4)
• use algebra tiles (Unit 3.1)
• use the guess-and-check method (esp when you have a coefficient in front of the $x^2-term$ (Unit 3.2)
• complete the square (Unit 3.5)
• use the quadratic formula (This section: Unit 3.6)
• The following conditions MUST be met before you can use the quadratic formula to find the solutions of an equation:
• There can be no term whose degree is higher than 2.
• The coefficient of the $x^2-term$ can’t be zero.
• One side of the equation must be zero.
• To begin solving with the quadratic equation, simplify the equation and convert it into standard form: $ax^2+bx+c=0$.

• Remember, the square root of a number has both, a positive and a negative answer.
• Ex: $\sqrt{16}=\pm4$ because $4^2=16$ and $(-4)^2=16$.
Discriminants and Complex Numbers
Roots
• To Find the Roots of a Polynomial’s Graph:
1. Start with the equation for a graphed polynomial in this form y = ax2 + bx + c.
2. Set y equal to zero.
3. Use the quadratic formula to solve for x.
4. Write each solution as a point (x, 0).
5. Those two points are the roots of the polynomial!