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To solve quadratic formulas To solve quadratic formulas there are 3 basic methods To solve quadratic formulas there are 3 basic methods 1. Factoring To solve quadratic formulas there are 3 basic methods 1. Factoring 2. Quadratic Formula To solve quadratic formulas there are 3 basic methods 1. Factoring 3. Completing the square 2. Quadratic Formula Completing the square involves several steps Completing the square involves several steps 1. get c by itself Completing the square involves several steps 1. get c by itself 2. Divide by a Completing the square involves several steps 1. get c by itself 2. Divide by a 3. Add (b/2)2 to both sides Completing the square involves several steps 1. get c by itself 2. Divide by a 4. Factor (put in exponential form) 3. Add (b/2)2 to both sides Completing the square involves several steps 1. get c by itself 2. Divide by a 4. Factor (put in exponential form) 3. Add (b/2)2 to both sides 5. Solve Completing the square involves several steps 1. 2. 3. 4. 5. get c by itself ? Divide by a ? Factor (put in exponential form) ? Add (b/2)2 to both sides ? Solve ? Let's do an example Let's do an example 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example x2 - 8x -20 = 0 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve a = 1 Let's do an example x2 - 8x -20 = 0 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve a = 1 Let's do an example x2 - 8x -20 = 0 b = -8 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve a = 1 Let's do an example x2 - 8x -20 = 0 b = -8 c = -20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve a ? 3x2 -5x +7 = 0 b ? c ? -2x2 +8x -4 = 0 c = b = a = Let's do an example First we get c by itself x2 - 8x -20 = 0 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example First we get c by itself x2 - 8x -20 = 0 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example First we get c by itself x2 - 8x -20 = 0 x2 - 8x = 20 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Now divide everything by a x2 - 8x -20 = 0 x2 - 8x = 20 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Now divide everything by a but since a is 1, we divide everything by 1 x2 - 8x -20 = 0 x2 - 8x = 20 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Now divide everything by a but since a is 1, we divide everything by 1 x2 - 8x -20 = 0 which means the equation doesn't change x2 - 8x = 20 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Then we add (b/2)2 to both sides x2 - 8x -20 = 0 x2 - 8x = 20 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Then we add (b/2)2 to both sides Since b = -8, then we will add 16 to both sides x2 - 8x -20 = 0 x2 - 8x = 20 +20 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Then we add (b/2)2 to both sides Since b = -8, then we will add 16 to both sides x2 - 8x -20 = 0 x2 - 8x = 20 +20 +16 +16 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example Then we add (b/2)2 to both sides Since b = -8, then we will add 16 to both sides x2 - 8x -20 = 0 x2 - 8x +16 = 36 x2 - 8x = 20 +20 +16 +16 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Then, factor the left side Let's do an example x2 - 8x -20 = 0 x2 - 8x +16 = 36 x2 - 8x = 20 +20 +16 +16 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Then, factor the left side Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 x2 - 8x = 20 +20 +16 +16 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Then, factor the left side Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 put in exponential form x2 - 8x = 20 +20 +16 +16 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Then, factor the left side Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 put in exponential form x2 - 8x = 20 (x - 4)2 = 36 +20 +16 +16 +20 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 x2 - 8x = 20 (x - 4)2 = 36 +20 +16 +16 +20 Last, solve the equation 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 +20 +16 √ +16 +20 Last, solve the equation First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +20 +16 √ +16 +20 Last, solve the equation First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +4 +20 +16 √ +4 +16 +20 Last, solve the equation First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +4 +20 +16 √ +4 +16 +20 x = +6 + 4 Last, solve the equation First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +4 +20 +16 √ +4 +16 +20 x = +6 + 4 Last, solve the equation x = 10 First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +4 +20 +16 √ +4 +16 +20 x = +6 + 4 Last, solve the equation x = 10 First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides x = -6 + 4 Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +4 +20 +16 √ +4 +16 +20 x = +6 + 4 Last, solve the equation x = 10 First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides x = -6 + 4 x = -2 Let's do an example (x - 4)(x - 4) = 36 x2 - 8x -20 = 0 x2 - 8x +16 = 36 √ x2 - 8x = 20 (x - 4)2 = 36 x - 4 = ±6 +4 +20 +16 √ +4 +16 +20 x = +6 + 4 Last, solve the equation x = 10 First square root 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve both sides x = -6 + 4 x = -2 Let's try a harder one 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 First get c by itself 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 +5 +5 First get c by itself 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 2x2 + 3x = 5 +5 +5 First get c by itself 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 2x2 + 3x = 5 +5 +5 Then divide by a 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 2x2 + 3x = 5 2 +5 2 +5 2 Then divide by a 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 2 3 2 +5 2 2 5 +5 2 Then divide by a 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 2 3 2 +5 2 2 5 +5 2 Now add (b/2)2 to both sides 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 2 16 3 2 9 +5 + 2 2 16 5 9 +5 2 Now add (b/2)2 to both sides 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 Now add (b/2)2 to both sides 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 Now we factor 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 Now we factor Unfortunately the fractions make it hard to factor 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 So we will just set it up like this (x )2 = 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 Use the first sign (x )2 = + 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 Square root (x )2 = + 3 4 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 9 16 49 x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 (x )2 = + 3 4 1. Get c by itself 4. Factor (expontential form) 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 Then we solve x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 √ (x )2 = + 3 4 1. Get c by itself 4. Factor (expontential form) √ 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 Then we solve x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 √ (x )2 = x + = ± + 3 4 4 3 1. Get c by itself 4. Factor (expontential form) √ 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 7 4 x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 x = - √ (x )2 = x + = ± + 4 3 3 4 4 3 + 7 4 1. Get c by itself 4. Factor (expontential form) √ 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 x = - 7 4 4 3 - 7 4 x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 x = - √ (x )2 = x = 1 x + = ± + 4 3 3 4 4 3 + 7 4 1. Get c by itself 4. Factor (expontential form) √ 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 x = - 7 4 x = -2.5 4 3 - 7 4 x2 + x + = Let's try a harder one 2x2 + 3x -5 = 0 x2 + x = 2x2 + 3x = 5 + 3 2 2 16 3 2 9 +5 + 16 2 9 2 16 5 9 +5 2 16 49 x = - √ (x )2 = x = 1 x + = ± + 4 3 3 4 4 3 + 7 4 1. Get c by itself 4. Factor (expontential form) √ 2. Divide by a 3. Add (b/2)2 to both sides 5. Solve 16 49 x = - 7 4 x = -2.5 4 3 - 7 4 |