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Methods for Solving Quadratic Equations Quadratics equations are of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0 Quadratics may have two, one, or zero real solutions. 1. FACTORING Leading Coefficient 𝒂 = 𝟏 Set the equation equal to zero. If the quadratic side is factorable, factor, then set each factor equal to zero. Example: 𝒙𝟐 = − 𝟓𝒙 − 𝟔 Move all terms to one side. 𝒙𝟐 + 𝟓𝒙 + 𝟔 = 𝟎 Factor. (𝒙 + 𝟑)(𝒙 + 𝟐) = 𝟎 Set each factor to zero and solve. 𝒙 + 𝟑 = 𝟎 𝒙 + 𝟐 = 𝟎 𝒙 = −𝟑 𝒙 = −𝟐 Leading Coefficient 𝒂 ≠ 𝟏 Set the equation equal to zero. Example: Move all terms to one side Multiply (𝑎 • 𝑐) Find the factors of (𝑎𝑐) that add to 𝑏 𝟐 𝟔𝒙 + 𝒙 = 𝟐 𝟔𝒙𝟐 + 𝒙 − 𝟐 = 𝟎 𝟔 • −𝟐 = −𝟏𝟐 𝟏 𝟐 −𝟑 𝟏𝟐 𝟔 𝟒 Replace 𝑏𝑥 with the factors of (𝑎𝑏). Group the first two terms from the last two. Factor the first group (find a common term). Factor the second group. There should be common term with each factor. Factor the common term Set each factor equal to zero. 𝟔𝒙𝟐 + 𝟒𝒙 − 𝟑𝒙 − 𝟐 = 𝟎 (𝟔𝒙𝟐 + 𝟒𝒙) + (−𝟑𝒙 − 𝟐) = 𝟎 (𝟐𝒙)(𝟑𝒙 + 𝟐) + (−𝟏)(𝟑𝒙 + 𝟐 ) = 𝟎 (𝟐𝒙)(𝟑𝒙 + 𝟐) + (−𝟏)(𝟑𝒙 + 𝟐) = 𝟎 Solve for x 𝒙=𝟐 ∗ 𝒂𝒃 + 𝒄𝒃 = (𝒃)(𝒂 + 𝒄) (𝟐𝒙 − 𝟏)(𝟑𝒙 + 𝟐) = 𝟎 (𝟐𝒙 − 𝟏) = 𝟎 (𝟑𝒙 + 𝟐) = 𝟎 𝟏 𝟐 𝒙 = −𝟑 2. PRINCIPLE OF SQUARE ROOTS If the quadratic equation involves a SQUARE and a CONSTANT (no first degree term), position the square on one side and the constant on the other side. Then take the square root of both sides. (Remember, you cannot take the square root of a negative number, so if this process leads to taking the square root of a negative number, there are no real solutions.) Example 1: 𝒙𝟐 − 𝟏𝟔 = 𝟎 Move the constant to the right side. 𝒙𝟐 = 𝟏𝟔 √𝒙𝟐 = ±√𝟏𝟔 Take the square root of both sides. 𝒙=±𝟒 Example 2: Move the constant to the other side. Isolate the square (divide both sides by 2). 𝟐(𝒙 + 𝟑)𝟐 − 𝟏𝟒 = 𝟎 𝟐(𝒙 + 𝟑)𝟐 = 𝟏𝟒 Take the square root of both sides (𝒙 + 𝟑)𝟐 = 𝟕 √(𝒙 + 𝟑)𝟐 = ±√𝟕 (𝒙 + 𝟑) = ±√𝟕 Solve for x (Simplify if possible) 𝒙 = −𝟑 ± √𝟕 3. COMPLETING THE SQUARE If the quadratic equation is of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where a ≠ 0 and the quadratic expression is not factorable, try completing the square. Example: 𝟑𝒙𝟐 + 𝟏𝟖𝒙 − 𝟑𝟑 = 𝟎 **If 𝒂 ≠ 𝟏, divide all terms by a before moving on. 𝒙𝟐 + 𝟔𝒙 − 𝟏𝟏 = 𝟎 Move the constant to the right side. 𝑏 Divide 2. 𝒙𝟐 + 𝟔𝒙 = 𝟏𝟏 Find 𝟑𝟐 = 𝟗 Add 𝑏 2 (2 ) . 𝑏 2 (2) to 𝟔 𝟐 =𝟑 𝒙𝟐 + 𝟔𝒙 + 𝟗 = 𝟏𝟏 + 𝟗 both sides of the equation. Factor the left side of the equation. (𝒙 + 𝟑)(𝒙 + 𝟑) = 𝟐𝟎 **Shortcut** Simplify the left side. (𝒙 + 𝟑)𝟐 = 𝟐𝟎 (𝒙 + 𝟐) = −𝒄 + (𝟐) 𝒃 𝟐 √(𝒙 + 𝟑)𝟐 = ±√𝟐𝟎 Take the square root of both sides. (𝒙 + 𝟑) = ±√𝟐𝟎 Solve for x. 𝒙 = −𝟑 ± √𝟐𝟎 Simplify if possible. 𝒙 = −𝟑 ± 𝟐√𝟓 4. QUADRATIC FORMULA Any quadratic equation of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0 can be solved for both real and imaginary solutions using the quadratic formula: 𝒙= −𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄 𝟐𝒂 Example: 𝟐𝒙𝟐 − 𝟒𝒙 = 𝟑 Set the equation equal to zero. Plug in the values for 𝒂, 𝒃, 𝒄 into the quadratic formula. 𝟐𝒙𝟐 − 𝟒𝒙 − 𝟑 = 𝟎 𝒂 = 𝟐, 𝒃 = −𝟒, 𝒄 = −𝟑 Simplify and solve for x. 𝒙= −(−𝟒)±√(−𝟒)𝟐 −𝟒(𝟐)(−𝟑) 𝟐(𝟐) 𝒙= 𝟒±√𝟏𝟔+𝟐𝟒 𝟒 𝒙= 𝟒±√𝟒𝟎 𝟒 = 𝟒±𝟐√𝟏𝟎 𝟒 = 𝟐±𝟏√𝟏𝟎 𝟐 𝟐±√𝟏𝟎 𝒙= 𝟐 ** The Quadratic Formula Method Works for all Quadratics. ** Notes: Remember quadratics can have one, two, or zero real roots, but can also have complex roots. Complex Numbers: 𝒊 = √−𝟏 Example: 𝟑 ± √−𝟒𝟎 = 𝟑 ± √(−𝟏)(𝟒𝟎) = 𝟑 ± 𝟐𝒊√𝟏𝟎 𝒃 𝟐