Ex - Factor and solve quadratic equations when a equals 1

- WE WANT TO SOLVE THE GIVEN QUADRATIC EQUATIONS BY FACTORING. NOTICE ALL THE EQUATIONS ARE IN THE FORM AX SQUARED + BX + C = 0 WHEN A = 1 MEANING THE FIRST TERM IS X SQUARED. SO LOOKING AT OUR FIRST EXAMPLE, REMEMBER THE FIRST STEP IN FACTORING IS TO LOOK FOR A GREATEST COMMON FACTOR. THE ONLY COMMON FACTOR THESE TERMS SHARE IS 1. SO IF THIS DOES FACTOR IT'LL FACTOR INTO TWO BINOMIALS WHERE THE FIRST TERMS COME FROM THE FACTORS OF X SQUARED WHICH WOULD BE X AND X. AND THEN THE SECOND TERMS OF THE BINOMIAL FACTORS WOULD COME FROM THE FACTORS OF 8 THAT ADD TO 6 OR THE FACTORS OF C THAT ADD TO B. AND SINCE 2 x 4 IS = 8 AND 2 + 4 = 6; 2 AND 4 ARE THE WINNING FACTORS. AND SINCE THE 2 AND 4 ARE BOTH POSITIVE WE WOULD HAVE X + 2 x X + 4. NOW THAT WE HAVE THE TRINOMIAL FACTORED WE CAN USE THE ZERO PRODUCT PROPERTY TO SOLVE THIS EQUATION. IF THIS PRODUCT IS EQUAL TO 0 THEN X + 2 MUST = 0 OR THE FACTOR X + 4 MUST = 0. AND NOW WE SOLVED THESE TWO EQUATIONS FOR X. SO WE'LL SUBTRACT 2 ON BOTH SIDES HERE. WE HAVE X = -2 OR SUBTRACT 4 ON BOTH SIDES HERE, WE HAVE X = -4. THESE WOULD BE OUR TWO SOLUTIONS TO THE QUADRATIC EQUATION. NOW LOOKING AT THE SECOND EXAMPLE, AGAIN THE ONLY COMMON FACTOR BETWEEN THESE THREE TERMS IS 1. SO IF THIS IS GOING TO FACTOR IT'LL FACTOR INTO TWO BINOMIALS. THE FIRST TERMS COME FROM THE FACTORS OF X SQUARED, THAT WOULD BE X AND X. AND THE SECOND TERMS WILL BE THE FACTORS OF -50 THAT ADD TO 23. AND SINCE -2 x 25 = -50 AND -2 + 25 DOES = 23 THE TWO FACTORS WE NEED ARE -2 AND 25. SO ONE FACTOR WILL BE X - 2. THE OTHER FACTOR WILL BE X + 25. AND THAT'LL SOLVE SINCE THIS PRODUCT = 0 EITHER X - 2 MUST = 0 OR X + 25 MUST = 0. SO HERE WE'LL ADD 2 TO BOTH SIDES SO IF X = 2 OR SUBTRACT 25 ON BOTH SIDES WE HAVE X = -25. NEXT EXAMPLE, THE ONLY COMMON FACTOR IS 1 SO AGAIN IF THIS DOES FACTOR, IT'LL FACTOR INTO TWO BINOMIALS. THE FIRST TERM OF EACH FACTOR WILL COME FROM THE FACTORS OF X SQUARED, THAT'S X AND X. THE SECOND TERMS WILL COME FROM THE FACTORS OF 15 THAT ADD TO -8. AND SINCE THE SUM OF THE FACTORS MUST BE NEGATIVE, BUT THE PRODUCT MUST BE POSITIVE WE'LL HAVE TO USE TWO NEGATIVE FACTORS OF 15. FOR EXAMPLE, -3 x -5 IS = 15, BUT -3 + -5 DOES GIVE US -8, THE COEFFICIENT IN THE MIDDLE TERM. SO OUR TWO FACTORS WILL BE X - 3 AND X - 5. AND NOW TO SOLVE THIS EQUATION, SINCE THIS PRODUCT IS EQUAL TO 0 EITHER X - 3 MUST = 0 OR X - 5 MUST = 0. SO HERE WE'LL ADD 3 TO BOTH SIDES OF THE EQUATION. WE HAVE X = 3 OR ADD 5 TO BOTH SIDES AND WE HAVE X = 5. LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE EXAMPLE. THERE ARE NO COMMON FACTORS SO THIS WILL FACTOR INTO TWO BINOMIALS. THE FIRST TERMS WILL COME FROM THE FACTORS OF X SQUARED. SO WE HAVE X AND X AND NOW WE WANT THE FACTORS OF -24 THAT ADD TO -5. WELL, LET'S SEE -3 x 8 WOULD BE -24, BUT THE SUM WOULD BE 5. WE WANT A -5 SO IF WE USE A -8 AND A 3 THIS WOULD GIVE US -24 AND THE SUM WOULD BE -5. SO THESE ARE THE TWO WINNING FACTORS. SO WE HAVE X - 8 x X + 3. THEREFORE, THE SOLUTIONS OCCUR WHEN X - 8 = 0 OR X + 3 = 0, ADD 8 TO BOTH SIDES. HERE WE HAVE X = 8, SUBTRACT 3 ON BOTH SIDES AND HERE WE HAVE X = -3.   OKAY, I HOPE YOU FOUND THESE EXAMPLES HELPFUL.