Skip to content
Quick Guides

Algebra I – Quadratic Equations

Preview 10 of 33 pages

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

  1. Page 1
    Read page 1 as text

    Study Guide Algebra I – Quadra Ɵ c Equa Ɵ ons 1. Solving Quadra Ɵ c Equa Ɵ ons What Is a Quadratic Equation? A quadratic equation is an equation that can be written in this standard form: Here: • ( a ) , ( b ) , and ( c ) are numbers, • ( x ) is the variable, • and (a ≠ 0) (because if ( a = 0 ) , the equation would not be quadratic anymore). Quadratic equations always include an ( x 2 ) term. That squared term is what makes them different from linear equations. Ways to Solve a Quadratic Equation There are three main methods for solving quadratic equations: 1. Factoring 2. Using the quadratic formula 3. Completing the square In this section, we’ll focus on factoring . Solving by Factoring Factoring is often the quickest method — when the equation can be factored easily. Steps for Factoring Follow these steps carefully:

  2. Page 2
    Read page 2 as text

    Study Guide 1. Move all terms to one side of the equation so that the other side equals 0. 2. Factor the expression. 3. Set each factor equal to zero. 4. Solve the resulting equations. 5. Check your answers by plugging them back into the original equation. Let’s see how this works in an example. Example 1 Solve: [x 2 - 6x = 16] Step 1: Move all terms to one side Subtract 16 from both sides so the equation equals 0: Step 2: Factor Now factor the quadratic expression: Step 3: Set each factor equal to zero Step 4: Solve each equation

  3. Page 3
    Read page 3 as text

    Study Guide So we have two possible solutions : Step 5: Check Your Answers Always substitute your answers back into the original equation to make sure they work. Original equation: Check ( x = 8 ) Check ( x = - 2 ) Final Answer The solutions are: Example 2 Solve: [y 2 = - 6y - 5] Step 1: Move all terms to one side

  4. Page 4
    Read page 4 as text

    Study Guide To factor, we need the equation to equal 0. Add (6y) and (5) to both sides: Step 2: Factor Now factor the quadratic expression: Step 3: Set each factor equal to zero Step 4: Solve So the solutions are: Step 5: Check Your Answers Substitute each value into the original equation: Original equation: Check ( y = - 5 )

  5. Page 5
    Read page 5 as text

    Study Guide Left side: Right side: 25 = 25 (Correct) Check ( y = - 1 ) Left side: Right side: 1 = 1 (Correct) Both values work! Example 3 Solve: [x 2 - 16 = 0] Step 1: Factor This is a difference of squares . So we write:

  6. Page 6
    Read page 6 as text

    Study Guide Step 2: Set each factor equal to zero Step 3: Solve So the solutions are: Step 4: Check Substitute into the original equation ( x 2 - 16 = 0 ) . For ( x = - 4 ) : For ( x = 4 ) : Both solutions satisfy the equation. Example 4 Solve: [x 2 + 6x = 0]

  7. Page 7
    Read page 7 as text

    Study Guide Step 1: Factor Here, there is no constant term . This is an incomplete quadratic. Factor out the greatest common factor (GCF), which is ( x ) : Step 2: Set each factor equal to zero or Step 3: Solve So the solutions are: Step 4: Check For ( x = 0 ) : For ( x = - 6 ) :

  8. Page 8
    Read page 8 as text

    Study Guide Both solutions are correct. Example 5 Solve: Step 1: Move Everything to One Side To solve by factoring, the equation must equal 0 . Subtract (x 2 + 6x - 5 ) from both sides and combine like terms: Now simplify: Step 2: Factor This is a perfect square trinomial : or written more simply,

  9. Page 9
    Read page 9 as text

    Study Guide Step 3: Solve This equation has one solution . (It is sometimes called a repeated or double root.) Step 4: Check Your Answer Substitute ( x = 2 ) into the original equation: Simplify both sides: Left side: Right side: The Quadratic Formula Factoring works well when the quadratic can be factored easily . However, many quadratic equations cannot be solved by factoring. This is especially true when the solutions are not rational numbers . In those cases, we use the quadratic formula :

  10. Page 10
    Read page 10 as text

    Study Guide Understanding the Formula The quadratic formula is used when the equation is written in standard form : Where: • ( a ) is the number in front of ( x 2 ) • ( b ) is the number in front of ( x ) • ( c ) is the constant (the number without a variable) The Discriminant Inside the square root is an important expression: This part is called the discriminant . The discriminant tells you how many real solutions the equation has. There are three possible cases : 1. If ( b 2 - 4ac > 0 ) → The equation has two different real roots . 2. If ( b 2 - 4ac = 0 ) → The equation has one real root (a repeated root). 3. If ( b 2 - 4ac < 0 ) → The equation has no real roots . Example 6 Solve for (x):

23 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.