Using the Quadratic Formula

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Examples, solutions, and videos to help Algebra I students learn how to use the quadratic formula to solve quadratic equations that cannot be easily factored.
Students understand that the discriminant, b² - 4ac, , can be used to determine whether a quadratic equation has one, two, or no real solutions.

New York State Common Core Math Algebra I, Module 4, Lesson 15

Worksheets for Algebra I, Module 4, Lesson 15 (pdf)

Lesson 15 Summary

You can use the sign of the discriminant, b² - 4ac, to determine the number of real solutions to a quadratic equation in the form ax² + bx + c = 0, where a ≠ 0.
If the equation has a positive discriminant, there are two real solutions.
A negative discriminant yields no real solutions and a discriminant equal to zero yields only one real solution.

Exercises 1–5
Solve the following equations using the quadratic formula.
1. x² - 2x + 1 = 0
2. 3b² + 4b + 8 = 0
3. 2t² + 7t - 4 = 0
4. q² - 2q - 1 = 0
5. m² - 4 = 3

Exercises 6–10
For Exercises 6–9, without solving, determine the number of real solutions for each quadratic equation.
6. p² + 7p + 33 = 8 - 3p
7. 7x² + 2x + 5 = 0
8. 2y² + 10y = y² + 4y - 3
9. 4z² + 9 = -4z

10. State whether the discriminant of each quadratic equation is positive, negative, or equal to zero on the line below the graph. Then identify which graph matches the discriminants below:

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