Grade 11 · Free math lesson

Advanced Algebra

After this lesson, you will be able to solve quadratic equations using three different methods.

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Skills in this unit

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Standards covered:A-APR.A.1A-APR.D.6A-APR.B.2A-APR.D.7A-REI.A.2N-RN.A.1N-RN.A.2F-LE.A.2F-IF.C.8.bF-LE.A.4F-BF.B.5N-CN.A.1N-CN.A.2N-CN.C.7F-BF.A.1.cF-BF.B.4

What Is a Quadratic?

A quadratic equation always has an x² term in it. It looks like ax² + bx + c = 0, where a, b, and c are numbers.

Example: x² + 5x + 6 = 0 is a quadratic. Here a=1, b=5, c=6.

Remember: Spot the x² — if it's there, it's quadratic!

Solve by Factoring

You split the equation into two brackets that multiply to zero. If two things multiply to zero, at least one of them must be zero.

Example: x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2 or x = -3.

Remember: Find two numbers that MULTIPLY to c and ADD to b.

Use the Quadratic Formula

When factoring is too hard, plug a, b, and c into this formula: x = (-b ± √(b²- 4ac)) / 2a. It always works!

Example: For x² + 5x + 6 = 0: x = (-5 ± √(25-24)) / 2 = (-5 ± 1) / 2 → x = -2 or x = -3.

Remember: Remember the formula as: Negative b, plus or minus the square root, all over 2a.

Worked example

Solve x² - 7x + 10 = 0 using factoring.

  1. 1Step 1: Find two numbers that multiply to 10 and add to -7. Those numbers are -2 and -5.
  2. 2Step 2: Write the factored form: (x - 2)(x - 5) = 0.
  3. 3Step 3: Set each bracket to zero — x - 2 = 0 gives x = 2, and x - 5 = 0 gives x = 5.

Answer: x = 2 or x = 5

The #1 mistake to avoid

Forgetting the ± in the quadratic formula — always check BOTH the + answer and the - answer, or you will miss one solution!

Key takeaways

  • A quadratic equation always contains an x² term and equals zero.
  • Factoring works by finding two numbers that multiply to c and add to b.
  • The quadratic formula x = (-b ± √(b²-4ac)) / 2a always works when factoring is tricky.

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