Algebra Know-It-ALL
Tables 22-1 through 22-5 show how we can convert the above equations to polynomial standard form. Note the last step in Table 22 ...
366 Quadratic Equations with Real Roots Are you confused? Suppose you see an equation in one variable. You think it’s a quadrati ...
which reduces to 7x^2 /2. This allows us to rewrite the right side of the original equation to obtain 7 x^2 /2+ 7 x− 5 = 7 x^2 / ...
368 Quadratic Equations with Real Roots The fact that the above general equation is a quadratic might not appear obvious, but wh ...
where neither p nor r equals 0. If either factor (that is, either binomial) happens to equal 0, then the entire expression on th ...
370 Quadratic Equations with Real Roots Solution The process of putting a quadratic into binomial factor form is called, not sur ...
That’s the left side of the original quadratic. Now we know that it can be written as (x+ 3)(x− 5) = 0 The roots are easy to fin ...
372 Quadratic Equations with Real Roots Such a root is said to have multiplicity 2. If the above polynomial expressions are plac ...
The left sides can be factored exactly as before, getting (x+ 1)^2 = 1 (x− 1)^2 = 4 (x+ 2)^2 = 16 (3x+ 2)^2 = 25 If we take the ...
374 Quadratic Equations with Real Roots x^2 + 4 x+ 4 = 16 9 x^2 + 12 x+ 4 = 25 Morphing into a perfect square When we come acros ...
We can take the square root of both sides and get x+ 1 =± 1 which can be expressed as the pair x+ 1 = 1 or x+ 1 =− 1 The solutio ...
376 Quadratic Equations with Real Roots Because a≠ 0 in any quadratic equation, we can divide each side by a, getting x^2 + (b/a ...
We’re not quite done yet, because this formula can be simplified. We have a common denominator, 2a, in the difference of the two ...
378 Quadratic Equations with Real Roots The discriminant In the second example above, b^2 − 4 ac= 0. In that case, it doesn’t ma ...
Solution We can work it the other way, multiplying the factors out to get the polynomial. Let’s rewrite the above squared binomi ...
380 Quadratic Equations with Real Roots Manipulate the following equation so it has the square of a binomial on the left side o ...
381 CHAPTER 23 Quadratic Equations with Complex Roots Now that we know a little bit about what to expect when the discriminant o ...
382 Quadratic Equations with Complex Roots Let’s take an example. Suppose that we work out the discriminant d for a quadratic, a ...
In the general polynomial standard form, we have a= 3, b= 0, and c= 75. The discriminant is therefore d=b^2 − 4 ac = 02 − (4 × 3 ...
384 Quadratic Equations with Complex Roots which simplifies to ax^2 +c= 0 Subtracting c from each side gives us ax^2 =−c We can ...
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