Algebra Know-It-ALL
and b^2 = 32 = 9 Therefore, d=b^2 − 4 ac = 9 − 90 =− 81 This tells us that the roots of the quadratic are not real numbers. To f ...
386 Quadratic Equations with Complex Roots Here’s a challenge! Prove the second of the above statements. Solution Consider once ...
Imaginary Roots in Factors Now that we’ve found a way to solve quadratics that have real coefficients, a real constant, and a ne ...
388 Quadratic Equations with Complex Roots Multiplying through be 3 gives us 3 x^2 + 75 = 0 That’s the original equation. The ge ...
When we add j[(c/a)1/2] to each side, we get x+j[(c/a)1/2]= 0 The binomial factor form of the original quadratic is therefore {x ...
390 Quadratic Equations with Complex Roots which simplifies to 2 x^2 + 79 = 0 Here’s a challenge! Investigate what happens in th ...
Dividing through by (−a), we obtain x^2 =c/(−a) Because c> 0 and −a> 0, we know that c/(−a)> 0. We can take the positiv ...
392 Quadratic Equations with Complex Roots The roots are therefore x= [−b±j(|d|1/2)] / (2a) = (− 6 ±j8) / (2 × 5) = (− 6 ±j8) / ...
Are you confused? If you wonder how the two trinomials in the above situation can be multiplied out, remember the rule you deriv ...
394 Quadratic Equations with Complex Roots This makes the above factored equation look like this: [x− (p+jq)][x− (p−jq)]= 0 To a ...
and further to x^2 + (b/a)x+c/a= 0 Multiplying through by a, we obtain ax^2 +bx+c= 0 which is the original general quadratic in ...
396 CHAPTER 24 Graphs of Quadratic Functions When a quadratic equation is in polynomial standard form but the 0 on the right is ...
Parabola opens upward Figure 24-2 shows a generic graph of a quadratic function of x with two real zeros, which we call r and s. ...
398 Graphs of Quadratic Functions Whenever a parabola opens upward, it has a single point at which it “bottoms out.” This point ...
values when the parabola opens upward. When a parabola opens downward, it has a single point at which it “peaks.” This point is ...
400 Graphs of Quadratic Functions Are you confused? By now, you might ask, “How can we can tell whether the parabola for a parti ...
The roots here are easily seen as x= 1 or x= 2. That means r= 1 and s= 2. Because the parabola opens upward, we should look for ...
402 Graphs of Quadratic Functions to get a clear picture of the parabola based on their locations. But we can find the y-interce ...
The absolute minimum is at the point (r, 0). The parabola seems to “sit upon” the x axis. Parabola opens downward Figure 24-6 il ...
404 Graphs of Quadratic Functions The absolute maximum is at the point (r, 0). The graph looks as if the parabola “hangs from” t ...
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