Skip to content
Quick Guides

Algebra II - Quadratics in One Variable

Preview 10 of 65 pages

See the first 10 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    Study Guide Algebra II – Quadra Ɵ cs in One Variable 1. Quadra Ɵ c Equa Ɵ ons A quadratic equation is a type of algebraic equation that includes a variable raised to the power of 2 (that is, squared). The general or standard form of a quadratic equation is: Here is what each part means: • (x) is the variable you are solving for. • (a), (b), and (c) are numbers called coefficients or constants . • (a) cannot be zero , because if (a = 0), the (x 2 ) term disappears and the equation is no longer quadratic. Quadratic equations appear often in algebra, physics, engineering, and many real - world problems. Because of this, it is important to know different ways to solve them. Common Methods to Solve Quadratic Equations There are four main algebraic methods used to solve quadratic equations. Each method works best in different situations. 1. Factoring Factoring means rewriting the quadratic expression as a product of two simpler expressions. Example idea: Once factored, you can set each factor equal to zero to find the solutions.

  2. Page 2
    Read page 2 as text

    Study Guide 2. Square Root Property This method is useful when the equation can be written in the form: To solve it, take the square root of both sides. Example idea: 3. Completing the Square Completing the square rearranges the equation so one side becomes a perfect square. This method is helpful when factoring is difficult or impossible. It is also the idea used to derive the quadratic formula. 4. Quadratic Formula The quadratic formula is a universal method that works for any quadratic equation , even when the other methods do not work easily. Students often use this method when the equation cannot be factored. Summary • A quadratic equation has the form (ax 2 + bx + c = 0) where (a ≠ 0) . • The equation contains a squared variable ((x 2 )) , which makes it quadratic. • There are four main methods to solve quadratic equations: 1. Factoring 2. Square Root Property 3. Completing the Square

  3. Page 3
    Read page 3 as text

    Study Guide 2. Solving Quadra Ɵ c by Factoring Quadratic equations can often be solved by factoring . This method uses an important rule called the Zero Product Property . The Zero Product Property The Zero Product Property says: If two numbers multiply to give 0 , then at least one of them must be 0 . In math form: If ab = 0 , then a = 0 or b = 0 . This idea is the key to solving quadratic equations by factoring. Example 1: Problem Solve: (2x 2 = - 9x - 4) Let’s solve this step by step. Step 1: Move all terms to one side To factor a quadratic equation, we first need to write it in the form: Add (9x) and (4) to both sides: Step 2: Factor the quadratic Now we factor the expression:

  4. Page 4
    Read page 4 as text

    Study Guide It factors into: Step 3: Apply the Zero Product Property If the product of two expressions equals 0 , then one of them must be 0 . So we set each factor equal to zero: or Step 4: Solve each equation Solve the first equation: Now solve the second equation: Final Solutions These are the values of x that make the original equation true.

  5. Page 5
    Read page 5 as text

    Study Guide Summary • Factoring is a method used to solve quadratic equations. • It relies on the Zero Product Property : • If (ab = 0), then (a = 0) or (b = 0). • Steps to solve by factoring: 1. Move all terms to one side so the equation equals 0 . 2. Factor the quadratic expression. 3. Set each factor equal to 0 . 4. Solve the resulting equations. Example result: For (2x 2 + 9x + 4 = 0) Solutions are: 3 . Quiz: Solving Quadra Ɵ cs by Factoring 1. Question Solve the equation: (x 2 + 2x - 8 = 0) Answer Choices  - 4, 2  - 2, 4  - 8, 1

  6. Page 6
    Read page 6 as text

    Study Guide Correct Answer - 4, 2 Why This Is Correct Factor the quadratic: Set each factor equal to zero: • (x + 4 = 0 ⇒ x = - 4) • (x - 2 = 0 ⇒ x = 2) So the solutions are - 4 and 2 . 2. Question Solve the equation: (x 2 - 11x = - 30) Answer Choices  - 5, 6  - 6, 5  5, 6 Correct Answer 5, 6 Why This Is Correct Move ( - 30) to the left:

  7. Page 7
    Read page 7 as text

    Study Guide Factor: Solve: • (x = 5) • (x = 6) 3. Question Solve the equation: (2x 2 = 28 - 10x) Answer Choices  - 7, 4  - 7, 2  - 4, 7 Correct Answer - 7, 2 Why This Is Correct Move everything to one side: Divide by 2: Factor:

  8. Page 8
    Read page 8 as text

    Study Guide Solutions: • (x = - 7) • (x = 2) 4. Question Solve the equation: (4x 2 = 25) Answer Choices Correct Answer Why This Is Correct Divide by 4: Take the square root: So the solutions are ( - 5/2) and (5/2) .

  9. Page 9
    Read page 9 as text

    Study Guide 5. Question Solve the equation: (6x 2 + x - 2 = 0) Answer Choices Correct Answer Why This Is Correct Factor the quadratic: Set each factor equal to zero: • (3x + 2 = 0 ⇒ x = - (2/3 ) • (2x – 1 = 0 ⇒ x = (1/2 ) So the solutions are – (2/3 ) and (1/2 ) . 4 . Solving Quadra Ɵ cs by the Square Root Property Another useful way to solve some quadratic equations is by using the Square Root Property . This method works best when the equation contains a squared term by itself, such as (x 2 ) or ((x - a) 2 ). The Square Root Property The Square Root Property states: x 2 = c ⇒ x = ± √ {c} This means that if a squared number equals a value I , then the variable can be either the positive square root or the negative square root of I . • If I is positive , the equation has two real solutions . • If I is negative , the equation has two imaginary (complex) solutions .

  10. Page 10
    Read page 10 as text

    Study Guide Example 1 : Solve Each Equation Let’s solve each equation step by step. 1. (x 2 = 48) Take the square root of both sides. Simplify the square root: Answer: 2. (x 2 = - 16) Take the square root of both sides: Since the square root of a negative number involves (i): Answer: 3. (5x 2 – 45 = 0) First isolate (x 2 ). Add 45 to both sides:

55 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.