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Unit 11 · Quadratic Functions

Solving Quadratic Equations

Zero product property, isolating a square, and the quadratic formula are three tools. Always set the equation equal to zero before factoring.

Algebra I Regents
AI-A.REI.4AI-A.SSE.3

Notes

Choose a method

If there is no x term, isolate and square-root. If it factors nicely, factor. Otherwise use x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac tells you how many real roots.

Vocabulary

discriminant

b² − 4ac; determines the number of real solutions of a quadratic.

Example: If b² − 4ac = 0, there is one real solution (a double root).

Worked examples

Try the prompt first. The steps stay folded until you ask to see them.

Factor

Solve x² − 5x − 14 = 0.

Common errors

Watch for

Forgetting the ± when taking square roots

Both 5² and (−5)² equal 25.

Fix: Write ± immediately after taking square roots.

Explore with an applet

Guided practice

Submit each answer to check it. The key is not shown until you do.

Question 1 of 3Algebra I RegentsAI-A.APR.3Algebra I Regents · June 2024 · Q9Non-calculatorproficient

The zeros of the function f(x) = x(x − 5)(3x + 6) are

Answer choices
Choose or type an answer first. The key stays hidden until you do.
Question 2 of 3Algebra I RegentsAI-A.REI.4Algebra I Regents · June 2023 · Q6Non-calculatoradvanced

When completing the square for x² − 18x + 77 = 0, which equation is a correct step in this process?

Answer choices
Choose or type an answer first. The key stays hidden until you do.
Question 3 of 3Algebra I RegentsAI-A.APR.3Algebra I Regents · June 2024 · Q9Non-calculatorproficient

The zeros of the function f(x) = x(x − 5)(3x + 6) are

Answer choices
Choose or type an answer first. The key stays hidden until you do.

Intervention

Practice factoring trinomials in isolation before putting them in equations.

Enrichment

A ball’s height h(t) = −16t² + 32t + 6. When does it hit the ground? Interpret the extra root.

Explain your reasoning

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  • I set the equation equal to zero so I could use…