Algebra 1 Unit 8: Quadratic Equations — Free Review Games.
This unit covers solving by factoring, quadratic formula and completing the square — essential concepts for Algebra 1. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers solving by factoring, quadratic formula and completing the square — essential concepts for Algebra 1. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Solving By Factoring
To solve a quadratic by factoring, set the equation equal to zero, factor the trinomial into two binomials, then apply the Zero Product Property. The Zero Product Property states that if ab = 0, then a = 0 or b = 0. Exams will test your ability to recognize factorable quadratics and correctly identify both solutions.
Key Points
- Always move all terms to one side so the equation equals zero before factoring
- Factor out a GCF first if one exists, then factor the remaining trinomial
- Set each factor equal to zero separately to find each solution
- Check solutions by substituting back into the original equation
Solve: x² + 5x + 6 = 0
Factor the trinomial: find two numbers that multiply to 6 and add to 5, which are 2 and 3, giving (x + 2)(x + 3) = 0. Set each factor equal to zero: x + 2 = 0 → x = −2, and x + 3 = 0 → x = −3. The solutions are x = −2 and x = −3.
2 Quadratic Formula
The quadratic formula solves any quadratic equation ax² + bx + c = 0 using x = (−b ± √(b² − 4ac)) / 2a. The discriminant (b² − 4ac) tells you the number and type of solutions: positive means two real solutions, zero means one real solution, negative means no real solutions. Exams frequently require identifying a, b, and c correctly, especially when terms are missing or reordered.
Key Points
- Memorize the formula: x = (−b ± √(b² − 4ac)) / 2a
- Rewrite the equation in standard form (ax² + bx + c = 0) before identifying a, b, c
- Discriminant > 0: two solutions; = 0: one solution; < 0: no real solutions
- Simplify the square root fully and reduce the fraction when possible
Solve: 2x² − 4x − 6 = 0
Identify a = 2, b = −4, c = −6, then substitute: x = (−(−4) ± √((−4)² − 4(2)(−6))) / (2·2) = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4. This gives x = (4 + 8)/4 = 3 or x = (4 − 8)/4 = −1.
3 Completing The Square
Completing the square rewrites a quadratic in vertex form by creating a perfect square trinomial on one side of the equation. This method is tested both as a solving technique and as the algebraic basis for deriving the quadratic formula. Exams commonly ask you to complete the square when the leading coefficient is 1 and when converting to vertex form y = a(x − h)² + k.
Key Points
- Move the constant to the right side, then add (b/2)² to both sides
- The left side becomes a perfect square trinomial: (x + b/2)²
- Take the square root of both sides (include ± on the right), then solve for x
- If a ≠ 1, divide every term by a before completing the square
Solve: x² + 8x − 9 = 0 by completing the square
Move the constant: x² + 8x = 9. Take half of 8, square it: (8/2)² = 16, and add to both sides: x² + 8x + 16 = 25. Factor the left side: (x + 4)² = 25. Take the square root: x + 4 = ±5, so x = 1 or x = −9.
Questions, answered.
What is Quadratic Equations?
Quadratic Equations is Unit 8 of Algebra 1, covering solving by factoring, quadratic formula and completing the square.
How to study for Algebra 1 Unit 8?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.