Algebra Know-It-ALL
This can be rearranged to get ax^2 +bx+ (c−k)= 0 which we can solve with the quadratic formula. The roots of this quadratic, whi ...
406 Graphs of Quadratic Functions To find the x-values of the points where the line crosses the parabola, we have a new quadrati ...
Our goal is to find the coordinates of points so we can draw an approximation of the parabola. We can’t get perfection with penc ...
408 Graphs of Quadratic Functions another number q somewhat larger than xmin. We can choose integers for these numbers to make t ...
Parabola opens downward Figure 24-9 is another generic graph of a quadratic function of x with no real zeros. Again, the parabol ...
410 Graphs of Quadratic Functions Are you confused? We’ve made a substantial claim: The x-value of the extremum for a quadratic ...
They-value at the absolute minimum point is ymin= 2 xmin^2 + 4 xmin+ 3 = 2 × (−1)^2 + 4 × (−1)+ 3 = 2 − 4 + 3 =− 2 + 3 = 1 From ...
412 Graphs of Quadratic Functions y= 2 x^2 + 4 x+ 3 = 2 × 12 + 4 × 1 + 3 = 2 + 4 + 3 = 9 The third point is therefore (1, 9). We ...
413 CHAPTER 25 Cubic Equations in Real Numbers Let’s move into single-variable cubic equations, also called cubics or third-degr ...
414 Cubic Equations in Real Numbers In the first case, the stand-alone constant is 0. If we want to be formal, we can write (x+ ...
Here’s a challenge! Solve the following cubic equation, and discover that the real-number root is an integer: (31/2x− 12 1/2)^3 ...
416 Cubic Equations in Real Numbers Binomial-factor form Here’s the general form of a cubic broken down into three binomial fact ...
Imagine that we were presented with the above equation for the first time, and we had never seen it in factored form. We could f ...
418 Cubic Equations in Real Numbers That’s a reasonable question. Dividing through by x can “knock the equation down” from third ...
and the solution set X is {0, −6,−8}. A new root shows up this time: x= 0! The fact that one of the roots is 0 caused us to inad ...
420 Cubic Equations in Real Numbers What are the real roots? The process of finding the real roots of a cubic in the binomial-tr ...
Subtracting 5 from each side and then dividing through by 3, we get x=−5/3. Next, we apply the qua- dratic formula to the trinom ...
422 Cubic Equations in Real Numbers Polynomial Equation of Third Degree Any single-variable cubic equation can be written so it ...
If we see a cubic in polynomial standard form and we find ourselves staring at it, para- lyzed with uncertainty, there’s some go ...
424 Cubic Equations in Real Numbers To set up the synthetic division “grid,” we write down the coefficients and the stand- alone ...
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