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Quadratic Formula and Equations Quadratic Formula and Equations

A quadratic equation is an equation of the second degree, meaning that for an equation in x, the greatest exponent on x is 2. Quadratics most commonly refer to vertically oriented parabolas—that is, parabolas that open upward or downward. The graph of a vertically oriented parabola has the shape of a rounded "v," and the bottom-most (or top-most) point is called the vertex. The equation for a parabola is usually written in either standard or vertex form; however, the standard form is more commonly used to solve for the x -intercepts, or roots. The standard form is y = ax 2 + bx + c for any real numbers a, b, c where a ≠ 0. The vertex form is y − k = a (x − b )2 with vertex (b, k ) and where a ≠ 0. Because x -intercepts are the points at which the graph crosses the x -axis, the solutions are always found by substituting 0 for y. The roots are often useful in solving real world problems, and there are three common ways to find the roots: factoring, using the quadratic formula, and completing the square. Not all quadratics can easily be factored, but if they can, the quickest way to solve them is to factor and use the zero product property. The zero product property basically states that if the product of two numbers is 0, then at least one of the numbers multiplied must be a 0. In other words, for any real numbers a and b, if ab = 0, then either a = 0 or b = 0. Consider a swimmer who starts at one end of a pool, swims down to pick up a ring at the bottom of the middle of the pool, and then surfaces at the other end of the pool with the ring. The equation y = x 2 − 6x + 9 can be used to model the path of the swimmer, where y is the water level in the pool measured in feet, and x is the time in seconds since the swimmer started. The equations below show how to solve for the roots of the equation to find the number of seconds it took the swimmer to reach the ring at the bottom of the pool: namely, by substituting 0 for y, factoring, and using the zero product property. 0 = x 2 − 6x + 9 Substitute 0 for y. 0 = [x − 3](x − 3) Factor x 2 − 6x + 9. Either [x − 3] = 0 or (x − 3) = 0 Use the zero product property. So x = 3 and x = 3 Solve each equation. In this example, the quadratic has only one repeated root, x = 3. This root is the time at which the swimmer reached the bottom of the pool. This quadratic can be graphed by substituting values into the equation to make a table of points, then graphing the realistic portion of the parabola, as shown below. The graph below illustrates that the parabola has a vertical line of symmetry that passes through (3, 0). The equation for the line of symmetry of this parabola is x = 3. The graph of the parabola continues infinitely; however, to model the path of the swimmer, only the points from 0 to 6 seconds are graphed. This is because the swimmer starts at x = 0 seconds, swims down for 3 seconds to get the ring, and then swims up for 3 seconds. Because not all quadratic equations can be factored, other methods for finding roots are needed. One other method of finding roots for a quadratic is to use the quadratic formula. In the formula, the plus or minus sign means to solve the formula twice—once with a plus, and once with a minus. In other words, given a quadratic equation in standard form, y = ax 2 + bx + c, the solutions can be found by Consider that a delayed space shuttle leaves Earth about 20 minutes after the scheduled departure. At 6 miles out, the shuttle turns around and returns to Earth. The distance of the shuttle from Earth can be described by the equation y − 6 = −0.1(x − 30)2, where x is the number of minutes the shuttle is in flight. The equations below show how to find the total number of minutes the shuttle was off the ground. To find the roots of the equation, first solve for standard form and substitute 0 for y, as shown in Step One. The resulting trinomial cannot easily be factored into two binomials, so the quadratic equation must be used to solve for the roots, as shown in Step Two. Step One Step Two To graph the parabola, plot and connect the two roots and the vertex. (The equation was originally given in vertex form.) If needed, more points can be found by substituting values for x into the equation. To graph the realistic portion of the parabola, graph only the portion in Quadrant I (see below). The original problem said that the shuttle was delayed by about 20 minutes. This is the first intercept, x ≈ 22.25 minutes. The vertex represents the point at which the shuttle was 6 miles from Earth. The second intercept, x ≈ 37.75 represents the time at which the shuttle returned to Earth. To find the total number of minutes the shuttle was in flight, subtract its liftoff and landing times. The shuttle was in flight for about 37.75 − 22.25 = 15.5 minutes. Some equations for parabolas may be solved more easily by completing the square. Completing the square forces the trinomial to be a perfect square by replacing the constant term with . In addition to finding roots, completing the square is also used for transforming an equation in standard form to vertex form. Furthermore, this method can be extended for use with the other conic sections . The quadratic formula can be derived by completing the square in y = ax 2 + bx + c. The equations below show how to solve for vertex form of y = x 2 – 6x + 7 and find its roots by completing the square. The strategy is to move the constant term opposite the trinomial and replace it with Then the new trinomial is written as the square of a binomial. In Step Two, the vertex form is y + 2 = (x − 3)2, and the vertex is (3 − 2). To find the roots, substitute 0 for y and solve for x. The two roots are and . To graph this parabola, the vertex and approximations for the roots can be plotted and connected. Step One Step Two b)2. Then The vertex can be found directly from vertex form, and it can also be found from standard form. From standard form, use b /2a to find the x -coordinate of the vertex and then substitute the result into the equation to find the y -coordinate of the vertex. The discriminant can be used to determine if the graph crosses the x -axis, and if so how many times. The discriminant is the expression under the radical in the quadratic formula, b 2 − 4ac. A square root usually yields two solutions, unless it is the square root of zero. The table summarizes the number and types of solutions that can occur and how they affect the appearance of the graph. The value of a in a quadratic equation also affects the placement of the graph on the plane. If a is positive, the graph opens upwards; if it is negative the graph opens downward. If a is greater than one, the graph will be narrow, and if a is a fraction between 0 and 1, the graph will be wide. This bit of information is especially useful because the value of a affects other types of graphs in the same ways as it does parabolas. All conic sections are quadratics because they have equations of the second degree. However, only the vertically oriented parabolas that have been summarized in this article are functions. Graphing calculators and computers perform functions by taking an input and giving an output. Hence, most graphing tools are only equipped to graph equations of functions. To graph a horizontally oriented parabola on a calculator, the graph must be broken into pieces that are functions. Then the equations for each piece are graphed on the same plane to create the appearance of one graph. see also Conic Sections; Functions and Equations; Graphs and Effects of Parameter Changes. Michelle R. Michael


From Yahoo Answers

Question:For the period 1990- 2001, the number of tickets sold (in millions) for Broadway road tours can be modeled by the function y= -10.4x^2 + 132x + 332 where x is the number of years since 1990. In what year were 750 million tickets sold for Broadway road tours?

Answers:y = number of tickets sold (in millions) So, just replace "y" by 750 (millions) : 750 = -10.4x^2 + 132x + 332 -10.4x^2 + 132x + 332 - 750 = 0 -10.4x^2 + 132x - 418 = 0 To find the roots for the quadratic equations : Roots = (-b +/- sqrt(b^2 - 4ac)) / 2a Where : a = -10.4 b = 132 c = -418 Verification of the rationality : sqrt(b^2 - 4ac) sqrt((132)^2 - (4*-10.4*-418)) sqrt(17424 - 17388.8) roots are rationals coz sqrt(b^2-4ac) = 5.933 ---> not negative root 1 = (-b + 5.933) / 2a root 1 = (-132 + 5.933) / (2 *-10.4) root 1 = -126.067 / -20.8 root 1 --------------------> 6.061 root 2 = (-b - 5.933) / 2a root 2 = (-132 - 5.933) / (2 *-10.4) root 2 = -137.933 / -20.8 root 2 --------------------> 6.631 The two roots are differents and give you between 6 and 7. That means you will reach 750 million tickets twice in the year 6 or 1996.

Question:problem: marketing research by a company has shown that the profit, P = in thousands of dollars, is related to the amount spent on advertising, x = in thousands of dollars, by the quadratic function P = f(x) = 230 + 20x - 0.5x^2 table: x: -20 , -10 , 0 , 10 , 20 , 30 , 40 , 50 P=f(x): -310 , -20 , 230 , 380 , 430 , 380 , 230 , -20 questions: A. what are the coordinates of the following points? Which have meaning in this context? Give the meaning or explain why the point has no meaning. 1. y-intercept 2. x-intercept 3. vertex 4. Point with the same y-value as the y-intercept B. when graphed what parts provide a good model of this company's profits. For what values of x does f(x) realistically model the company's profits. Explain. C. What advice do you have for this company as it considers its advertising budget?

Answers:1. y-intercept => let x= 0 f(x) = 230 + 20x - 0.5x^2 f(x) = 230 + 20(0) - 0.5(0)^2 f(x) = 230 y-intercept = 230 2. x-intercept => let y = 0 f(x) = 230 + 20x - 0.5x^2 0 = 230 + 20x - 0.5x^2 use formula to solve 3. vertex f(x) = 230 + 20x - 0.5x^2 f '(x) = 20 - 0.5x Max pt. 20 - 0.5x = 0 => x = 40 => y = 230 + 20(40) - 0.5(40)^2 4. Point with the same y-value as the y-intercept x = 230 + 20x - 0.5x^2 0 = 230 + 19x - 0.5x^2 use formula to solve QED

Question:thank you!! 1.The demand for a product is p=7000-2x dollars, and supply is given by q=.01x^2+2x+1000 dollars, where x is the number of units supplied or demanded. Find the number of units when supply equals demand. 2.The area (in square meters) of a rectangular parking lot is found by equation: y=250x-x^2, where x is equals to the length of the parking lot in meters. If the parking lot must be at least 8000 square meters, find possible values for x. 3.The area (in square meters) of a rectangular parking lot is found by the equation y=100x-x^2, where x is the length of the parking lot in meters. If the parking lot must be smaller than 500 square meters, find possible values for x. 4.An object is thrown upward, and its height in feet is given by s(t)=-16t+80t+3, where t is seconds after being thrown. a)What is the initial height of the object? b)After how many seconds does the object hit the ground? 5) An open box of volume 32 inch^3 is to be made from from a square piece of tin by cutting 2 inch squares, from each corner and turning upsides. Find the dimensions of the piece of tin.

Answers:1.) p = 7000 - 2x q = 0.01x^2 + 2x + 1000 >> Find the number of units when supply equals demand. << if p = q then 7000 - 2x = 0.01x^2 + 2x + 1000 0.01x^2 + 4x - 6000 = 0 <== quadratic 2.) y = 250x - x^2 = area >> If the parking lot must be at least 8000 square meters << 8000 = 250x - x^2 x^2 - 250x + 8000 = 0 <=== quadratic

Question:a. Suppose a parabola has a vertex in Quadrant IV and a < 0 in the equation y=ax^2+bx+c. How many real solutions will the equation ax^2+bx+c=0 have? b. A ball is thrown upward from ground level. Its height h, in feet, above the ground after t seconds is h = 48t - 16t^2. Find the maximum height of the ball. Any help would be appreciated Thank You.

Answers:a. no real solutions If a < 0, the parabola opens down. n this case, the vertex is in Q IV and the parabola opens down, so it will never cross the x-axis (where y = 0). b. By factoring the equation, you can rewrite it as h = -16t(t - 3) The zeros are h = 0 and h = 3 These show the times at which the ball would be on the ground. Because of symmetry, the highest point would occur halfway between these times, that is, 1.5 second. Substitute 1.5 for t and solve to find the height at that point. h = 36 feet

From Youtube

77 word problems with quadratic functions :WEBSITE: www.teachertube.com solving word problems with quadratic functions

Int Algebra: The Quadratic Function :www.mindbites.com This 84 minute intermediate algebra lesson focuses on the quadratic function y= ax^2 + bx + c, a 0. This lesson will help you understand and learn to: - work with the quadratic function of the form y= ax^2 + bx + c, a 0 to draw parabolas - graph y = ax^2 with different values for a - graph y = ax^2 + c with different values for c - complete the square on the form y = ax^2 + bx + c to change to the form y = a(xp)^2 +q in order to find the vertex, the equation of the axis of symmetry, the minimum or maximum value, the range of the function, and what happens when (p is positive? negative?) (q is positive? negative?) - solve minimum and maximum word problems - work with parabolas to determine if they are functions or not, if they are expanded vertically (stretch in the y direction), if they are compressed vertically (shrink in the y direction), and if they open upward or downward Thislesson contains explanations of the concepts and 16 example questions with step by step solutions plus 6 interactive review questions with solutions. Lessons that will help you with the fundamentals of this lesson: - 170 Factoring Polynomials (www.mindbites.com - 210 Relations & Functions (Domain, Range, Functional Notation, & Inverses of Functions) (www.mindbites.com