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MAT122 :: Lecture Note :: Week 12
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f(x) = ae^(rt)
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Although he may not always recognize his bondage,
modern man lives under a tyranny of numbers.
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A quadratic equation, in standard form, is a 2nd-degree polynomial set equal to zero.
ax^2 + bx + c = 0 where a, b, c are coefficients and a ≠ 0 [if a = 0, then it's a linear equation] a is the quadratic coefficient b is the linear coefficient c is the constant coefficientMany of us think "four" when we see the word "quad." For example, a quadrilateral is a four-sided polygon. Therefore, it is not obvious as to why quadratics are called quadratics. The word quadratus is Latin for "square" and the variable in the 1st term of a quadratic expression is squared. [Note: squares have four sides.]
Remembering the quadratic formula is non-trivial.
x = (-b +- sqrt(b^2 - 4ac)) / (2a) ...or... x = (-b +- (b^2 - 4ac)^(1/2)) / (2a)The expression inside the square-root is called the discriminant. According to Merriam-Webster online, a discriminant is a "mathematical expression providing a criterion for the behavior of another more complicated expression, relation, or set of relations."
The solutions to the equation are called the roots of the equation or the zeros of the function.
let d stand for discriminant if d > 0, then two distinct real roots if d = 0, then one distinct real root if d < 0, then two distince complex rootsThe graph of a quadratic function is a parabola whose major axis is parallel to the y-axis.
As
|a|increases, the parabola becomes narrower; as|a|decreases, the parabola becomes wider.If a, b, and c are real numbers and the domain of f is the set of real numbers, then the zeros of f are exactly the x-coordinates of the points where the graph touches the x-axis. It follows from the above that, if the discriminant is positive, the graph touches the x-axis at two points, if zero, the graph touches at one point, and if negative, the graph does not touch the x-axis.The turning point for a parabola is called its vertex.
given f(x) = ax^2 + bx + c vertex for f(x) is (-b/(2a), f(-b/(2a)))Every parabola has an axis of symmetry which is the line that runs down its center. This line divides the graph into two perfect halves and it passes through the vertex.
equation of the axis of symmetry: x = -b/(2a)MathBabbler is not sure about the following being the "simplest."
"Often, the simplest way to solve 'ax^2 + bx + c = 0' for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor."9x^2 - 6x = 0 3x(3x - 2) = 0 # factor out greatest common factor 3x 3x = 0 ... 3x - 2 = 0 # set each linear equation to zero x = 0 x = 2/3 # solve each linear equation for xThis might be useful when having to solve a problem by hand; however, once you have a program all you have to do is input the coefficients and the program will output the roots.
MathBabbler copied the following from the web.
"This formula is as important and widely used as the Pythagorean Theorem. Teachers do not have mercy on students who do not remember the quadratic formula, unless they can help themselves by completing the square instead!"Hmmm... completing-the-square...
multiply each term by the multiplicative inverse of the quadratic coefficient add 1/2 of the linear coefficient to both sides1728.com and Math.Lamar.edu and Dizzy over Lizzie and QuadraticEquation.org
From the DoD (Department of Duh)...
"Part of the problem is that we are less apt to remember things that we aren't doing on a regular basis. If you aren't using the Quadratic Equation each day, you are going to be less apt to remember it when you do need it. This is something that many people find to be true as time goes on."There are lots of "quadratic songs" on YouTube... The Quadratic Song
Books.Google.com::Homework Helpers: Algebra
Exercises
(1) Simple interest formula: V = P(1 + r)^t $5,000 invested for 2 years grows to $5,618. What was the simple interest rate?5618 = 5000(1 + r)^2 # perform substitutions 5618/5000 = (1 + r)^2 # divide both sides by 5000 1.1236 = (1 + r)^2 sqrt(1.1236) = sqrt((1 + r)^2) # square-root both sides 1.06 = 1 + r 1.06 - 1 = r # subtract 1 from both sides 0.06 = r 0.06 * 100(%) = 6% # convert into a percentage # re-do this problem using the quadratic formula 5618 = 5000(1 + r)^2 # perform substitutions 5618 = 5000(1 + r)(1 + r) # expand the exponent 5618 = 5000(1 + r + r + r^2) # FOIL 5618 = 5000(1 + 2r + r^2) # combine like terms 5618 = 5000 + 10000r + 5000r^2 # distribute the 5000 0 = -618 + 10000r + 5000r^2 # subtract 5618 from both sides a=5000 b=10000 c=-618 # identify a,b,c values # substitute into the quadratic formula and evaluate r = (-10000 + sqrt(10000^2 - 4(5000)(-618))) / (2(5000)) r = 0.06 0.06 * 100(%) = 6% r = (-10000 - sqrt(10000^2 - 4(5000)(-618))) / (2(5000)) r = -2.06 # real number solution is correct, but the negative answer # doesn't make sense in the context of this problem(2) A rectangle having an area of 52 cm^2 has a length that is 1 cm more than 3 times its width. What is its perimeter? [area = l * w; perimeter = 2l + 2w]A = l * w l = 3w + 1 # 1 more than 3 times width 52 = (3w + 1) * w # substitute into Area formula 52 = 3w^2 + w # distribute w on the rhs 0 = 3w^2 + w - 52 # subtract 52 from both sides a=3 b=1 c=-52 # identify a,b,c values # substitute into the quadratic formula and evaluate w = (-1 + sqrt(1^2 - 4(3)(-52))) / (2(3)) w = 4 w = (-1 - sqrt(1^2 - 4(3)(-52))) / (2(3)) w = -4.3 # negative width doesn't make any sense... solution is discarded l = 3(4) + 1 # compute the length using the width l = 13 # substitute length and width into Perimeter formula and evaluate P = 2l + 2w P = 2(13) + 2(4) P = 26 + 8 p = 34 cmEducation.Yahoo.com::Quadratic Equations
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Quadratic equation in standard form:
ax^2 + bx + c = 0ax^2 + bx + c = 0 [2nd-degree polynomial] where a, b, c are coefficients and a ≠ 0 [if a = 0, then it's a linear equation] a is the quadratic coefficient b is the linear coefficient c is the constant coefficientQuadratic equations can always be solved by using the quadratic formula.
x = (-b +- sqrt(b^2 - 4ac)) / (2a) b^2 - 4ac is called the discriminant (d) if d > 0, then two distinct real roots if d = 0, then one distinct real root if d < 0, then two distince complex rootsThe graph of a quadratic function is a parabola.
f(x) = ax2 + bx + c a > 0 implies the parabola opens up a < 0 implies the parabola opens down when |a| increases, the parabola becomes narrower when |a| decreases, the parabola becomes wider the high (a < 0) and low point (a > 0) of a parabola is its vertex x-coordinate of the vertex: x = -b/(2a) vertex ordered-pair: (x, f(x)) or (-b/(2a), f(-b/(2a))) vertical intercept: (0, c) axis of symmetry: x = -b/(2a) domain: (-INF, INF) range: if vertex is (h, k) [k, INF) when a > 0 and (-INF, k] when a < 0
Hmmm... completing-the-square...
1) multiply each term by the multiplicative inverse of the quadratic coefficient 2) add 1/2 of the linear coefficient to both sides f(x) = a(x - h)^2 + k, vertex is (h, k)
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