6.2 The Quadratic Formula

Transcription

1 . The Quadratic Formula Factoring and completing the square are great tools to use when solving quadratics. Unfortunately, as we saw in the last homework assignment, there are some fairly ugly looking answers when completing the square. Now imagine that you had a quadratic with negative fraction coefficients. Wouldn t it be easier if we just had a formula for quadratics that you could just plug into to get the answer? Oh, wait. We do. It s called the quadratic formula, and it says that if you have a quadratic 0, then: 4 The Quadratic Formula Origin Story Every good superhero has an origin story. Wolverine came from the Weapon X project, Superman was sent to Earth by his father, and Batman, well, Batman was just always awesome. The quadratic formula actually started off as completing the square. Let s start with the basic standard form quadratic and factor out the value so we can pretend 1 after that. 0 0 Now we can just deal with the quadratic in the parentheses so the value doesn t throw off the completing the square process. We ll just have to remember to put it back in at the end. So let s write that quadratic in as an equivalent expression in vertex form (i.e. complete the square). Add and subtract half the coefficient squared. Factor the trinomial. Get a common denominator. Add the last two terms

2 Whew! We can t forget about that we originally factored out. Let s put that back in Notice the quadratic is now in vertex form and that the vertex is at. Finally, we re ready to solve Subtract the fraction from both sides. Multiply through the negative. Divide both sides by. We must consider both square roots. Subtract the fraction from both sides. And that, my friends, is how the average, everyday completing the square turned into the superhero QUADRATIC FORMULA! Using the Quadratic Formula Let s take a factorable quadratic equation and use the quadratic formula. If we wanted to solve the equation 450, we know that we could factor and find This would mean that the solutions would be 5 and 1. Now let s confirm that the quadratic formula will give us the same solutions for

3 We got the same answers! The quadratic formula works! Now let s use the quadratic formula for an equation that may not factor We can double check that these are the correct solutions by substituting them back into the original equation. Plugging in 1 makes the statement true and so does plugging in. (The fraction just takes a bit more work!) The Parts of the Quadratic Formula The quadratic formula not only solves quadratics, but it also can give us information quickly. For example, we can see the coordinate of the vertex right away. Remember how earlier we noted that the quadratic formula came from completing the square and putting the quadratic in vertex form? Well, there s the coordinate of the vertex sitting right in the formula. And the plus or minus blah blah? That represents moving the same distance to either side of the vertex to find the zeros. Pretty cool, huh? A quadratic can have a no zeros, a single zero, or two zeros. There is a hint in the quadratic formula as to which one this will be. We know we can t take the square root of the negative, so if 4 is negative, we ll get no solution. That means there will be no zeros. That part of the formula, 4, is called the discriminant. If the discriminant is zero, then there will be a single zero. That s because we ll be adding and then subtracting zero which gives us the same value either way. If the discriminant is positive, then we ll have our regualar two zeros. We can also tell if the parabola opens up or down based on the value. If it s positive, then the parabola opens up. If is negative, then the parabola has been reflected vertically meaning it opens downward. What can you tell from this quadratic formula? There are no zeros since the discriminant is 111, the vertex is at, and the parabola opens down. We can also reconstruct the original quadratic equation as

4 Lesson. Solve the following quadratic equations using any method

5 Find the coordinate of the vertex and determine how many zeros the quadratic has