Step by step solution :
Step 1 :1.1 Evaluate : (4x-1)2 = 16x2-8x+1
Step 2 :Pulling out like terms :2.1 Pull out lượt thích factors:16x2 - 8x - 24=8•(2x2 - x - 3)Trying to factor by splitting the middle term
2.2Factoring 2x2 - x - 3 The first term is, 2x2 its coefficient is 2.The middle term is, -x its coefficient is -1.The last term, "the constant", is -3Step-1 : Multiply the coefficient of the first term by the constant 2•-3=-6Step-2 : Find two factors of -6 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, -3 and 22x2 - 3x+2x - 3Step-4 : add up the first 2 terms, pulling out like factors:x•(2x-3) địa chỉ up the last 2 terms, pulling out common factors:1•(2x-3) Step-5:Add up the four terms of step4:(x+1)•(2x-3)Which is the desired factorizationEquation at the end of step 2 :
8 • (2x - 3) • (x + 1) = 0
Step 3 :Theory - Roots of a product :3.1 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.
Equations which are never true:3.2Solve:8=0This equation has no solution. A a non-zero constant never equals zero.
Solving a Single Variable Equation:3.3Solve:2x-3 = 0Add 3 to lớn both sides of the equation:2x = 3 Divide both sides of the equation by 2:x = 3/2 = 1.500
Solving a Single Variable Equation:3.4Solve:x+1 = 0Subtract 1 from both sides of the equation:x = -1
Supplement : Solving Quadratic Equation DirectlySolving 2x2-x-3 = 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:4.1Find the Vertex ofy = 2x2-x-3Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up & accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,2, 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 model 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 to lớn be able to 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 0.2500Plugging into the parabola formula 0.2500 for x we can calculate the y-coordinate:y = 2.0 * 0.25 * 0.25 - 1.0 * 0.25 - 3.0 or y = -3.125Parabola, Graphing Vertex & X-Intercepts :
Root plot for : y = 2x2-x-3 Axis of Symmetry (dashed) x= 0.25 Vertex at x,y = 0.25,-3.12 x-Intercepts (Roots) : Root 1 at x,y = -1.00, 0.00 Root 2 at x,y = 1.50, 0.00Solve Quadratic Equation by Completing The Square
4.2Solving2x2-x-3 = 0 by Completing The Square.Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :x2-(1/2)x-(3/2) = 0Add 3/2 to both side of the equation : x2-(1/2)x = 3/2Now the clever bit: Take the coefficient of x, which is 1/2, divide by two, giving 1/4, and finally square it giving 1/16Add 1/16 khổng lồ both sides of the equation :On the right hand side we have:3/2+1/16The common denominator of the two fractions is 16Adding (24/16)+(1/16) gives 25/16So adding to lớn both sides we finally get:x2-(1/2)x+(1/16) = 25/16Adding 1/16 has completed the left hand side into a perfect square :x2-(1/2)x+(1/16)=(x-(1/4))•(x-(1/4))=(x-(1/4))2 Things which are equal to the same thing are also equal lớn one another. Sincex2-(1/2)x+(1/16) = 25/16 andx2-(1/2)x+(1/16) = (x-(1/4))2 then, according lớn the law of transitivity,(x-(1/4))2 = 25/16We"ll refer to lớn this Equation as Eq.
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#4.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(1/4))2 is(x-(1/4))2/2=(x-(1/4))1=x-(1/4)Now, applying the Square Root Principle lớn Eq.#4.2.1 we get:x-(1/4)= √ 25/16 add 1/4 to both sides to obtain:x = 1/4 + √ 25/16 Since a square root has two values, one positive & the other negativex2 - (1/2)x - (3/2) = 0has two solutions:x = 1/4 + √ 25/16 orx = 1/4 - √ 25/16 note that √ 25/16 can be written as√25 / √16which is 5 / 4
Solve Quadratic Equation using the Quadratic Formula
4.3Solving2x2-x-3 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B and C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 2B= -1C= -3 Accordingly,B2-4AC=1 - (-24) = 25Applying the quadratic formula : 1 ± √ 25 x=—————4Can √ 25 be simplified ?Yes!The prime factorization of 25is5•5 khổng lồ be able lớn remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Second root).√ 25 =√5•5 =±5 •√ 1 =±5 So now we are looking at:x=(1±5)/4Two real solutions:x =(1+√25)/4=(1+5)/4= 1.500 or:x =(1-√25)/4=(1-5)/4= -1.000