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Step by Step Solution
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Step by step solution :
Step 1 :
x Simplify — 5Equation at the kết thúc of step 1 : (x2) x (———— - —) - 6 = 0 5 5
Step 2 :x2 Simplify —— 5 Equation at the over of step 2 : x2 x (—— - —) - 6 = 0 5 5
Step 3 :Adding fractions which have a common denominator :3.1 Adding fractions which have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x2 - (x) x2 - x ———————— = —————— 5 5 Equation at the over of step 3 : (x2 - x) ———————— - 6 = 0 5
Step 4 :Rewriting the whole as an Equivalent Fraction :4.1Subtracting a whole from a fraction Rewrite the whole as a fraction using 5 as the denominator :
6 6 • 5 6 = — = ————— 1 5 Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction and the other fraction involved in the calculation mô tả the same denominator
Step 5 :Pulling out lượt thích terms :5.1 Pull out like factors:x2 - x=x•(x - 1)Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions showroom the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce to lớn lowest terms if possible:
x • (x-1) - (6 • 5) x2 - x - 30 ——————————————————— = ——————————— 5 5 Trying to factor by splitting the middle term5.3Factoring x2 - x - 30 The first term is, x2 its coefficient is 1.The middle term is, -x its coefficient is -1.The last term, "the constant", is -30Step-1 : Multiply the coefficient of the first term by the constant 1•-30=-30Step-2 : Find two factors of -30 whose sum equals the coefficient of the middle term, which is -1.
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Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -6 and 5x2 - 6x+5x - 30Step-4 : địa chỉ cửa hàng up the first 2 terms, pulling out like factors:x•(x-6) showroom up the last 2 terms, pulling out common factors:5•(x-6) Step-5:Add up the four terms of step4:(x+5)•(x-6)Which is the desired factorization
Equation at the kết thúc of step 5 : (x + 5) • (x - 6) ————————————————— = 0 5
Step 6 :When a fraction equals zero :6.1 When a fraction equals zero ...Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.Here"s how:
(x+5)•(x-6) ——————————— • 5 = 0 • 5 5 Now, on the left hand side, the 5 cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:(x+5) • (x-6)=0
Theory - Roots of a sản phẩm :6.2 A hàng hóa of several terms equals zero.When a product of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going to lớn solve as many equations as there are terms in the productAny solution of term = 0 solves sản phẩm = 0 as well.
Solving a Single Variable Equation:6.3Solve:x+5 = 0Subtract 5 from both sides of the equation:x = -5
Solving a Single Variable Equation:6.4Solve:x-6 = 0Add 6 khổng lồ both sides of the equation:x = 6
Supplement : Solving Quadratic Equation DirectlySolving x2-x-30 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square và by using the Quadratic Formula
Parabola, Finding the Vertex:7.1Find the Vertex ofy = x2-x-30Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up and accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,1, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can mã sản phẩm many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want khổng lồ be able khổng lồ find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is 0.5000Plugging into the parabola formula 0.5000 for x we can calculate the y-coordinate:y = 1.0 * 0.50 * 0.50 - 1.0 * 0.50 - 30.0 or y = -30.250
Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = x2-x-30 Axis of Symmetry (dashed) x= 0.50 Vertex at x,y = 0.50,-30.25 x-Intercepts (Roots) : Root 1 at x,y = -5.00, 0.00 Root 2 at x,y = 6.00, 0.00Solve Quadratic Equation by Completing The Square
7.2Solvingx2-x-30 = 0 by Completing The Square.Add 30 to lớn both side of the equation : x2-x = 30Now the clever bit: Take the coefficient of x, which is 1, divide by two, giving 1/2, & finally square it giving 1/4Add 1/4 to lớn both sides of the equation :On the right hand side we have:30+1/4or, (30/1)+(1/4)The common denominator of the two fractions is 4Adding (120/4)+(1/4) gives 121/4So adding to both sides we finally get:x2-x+(1/4) = 121/4Adding 1/4 has completed the left hand side into a perfect square :x2-x+(1/4)=(x-(1/2))•(x-(1/2))=(x-(1/2))2 Things which are equal lớn the same thing are also equal to one another. Sincex2-x+(1/4) = 121/4 andx2-x+(1/4) = (x-(1/2))2 then, according to the law of transitivity,(x-(1/2))2 = 121/4We"ll refer to lớn this Equation as Eq. #7.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(1/2))2 is(x-(1/2))2/2=(x-(1/2))1=x-(1/2)Now, applying the Square Root Principle lớn Eq.#7.2.1 we get:x-(1/2)= √ 121/4 Add 1/2 to both sides to obtain:x = một nửa + √ 121/4 Since a square root has two values, one positive và the other negativex2 - x - 30 = 0has two solutions:x = 1/2 + √ 121/4 orx = 50% - √ 121/4 note that √ 121/4 can be written as√121 / √4which is 11 / 2