Rearrange:Rearrange the equation by subtracting what is to lớn the right of the equal sign from both sides of the equation : x^2-x-10-(2)=0
Step by step solution :
Step 1 :Trying lớn factor by splitting the middle term
1.1Factoring x2-x-12 The first term is, x2 its coefficient is 1.The middle term is, -x its coefficient is -1.The last term, "the constant", is -12Step-1 : Multiply the coefficient of the first term by the constant 1•-12=-12Step-2 : Find two factors of -12 whose sum equals the coefficient of the middle term, which is -1.
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 & 3x2 - 4x+3x - 12Step-4 : địa chỉ cửa hàng up the first 2 terms, pulling out like factors:x•(x-4) showroom up the last 2 terms, pulling out common factors:3•(x-4) Step-5:Add up the four terms of step4:(x+3)•(x-4)Which is the desired factorizationEquation at the over of step 1 :
(x + 3) • (x - 4) = 0
Step 2 :Theory - Roots of a hàng hóa :2.1 A product of several terms equals zero.When a hàng hóa 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 product = 0 as well.
Solving a Single Variable Equation:2.2Solve:x+3 = 0Subtract 3 from both sides of the equation:x = -3
Solving a Single Variable Equation:2.3Solve:x-4 = 0Add 4 khổng lồ both sides of the equation:x = 4
Supplement : Solving Quadratic Equation DirectlySolving x2-x-12 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square & by using the Quadratic Formula
Parabola, Finding the Vertex:3.1Find the Vertex ofy = x2-x-12Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up và 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 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 lớn 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 - 12.0 or y = -12.250Parabola, Graphing Vertex và X-Intercepts :
Root plot for : y = x2-x-12 Axis of Symmetry (dashed) x= 0.50 Vertex at x,y = 0.50,-12.25 x-Intercepts (Roots) : Root 1 at x,y = -3.00, 0.00 Root 2 at x,y = 4.00, 0.00Solve Quadratic Equation by Completing The Square
3.2Solvingx2-x-12 = 0 by Completing The Square.Add 12 lớn both side of the equation : x2-x = 12Now the clever bit: Take the coefficient of x, which is 1, divide by two, giving 1/2, và finally square it giving 1/4Add 1/4 to lớn both sides of the equation :On the right hand side we have:12+1/4or, (12/1)+(1/4)The common denominator of the two fractions is 4Adding (48/4)+(1/4) gives 49/4So adding khổng lồ both sides we finally get:x2-x+(1/4) = 49/4Adding 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 khổng lồ the same thing are also equal khổng lồ one another. Sincex2-x+(1/4) = 49/4 andx2-x+(1/4) = (x-(1/2))2 then, according khổng lồ the law of transitivity,(x-(1/2))2 = 49/4We"ll refer to this Equation as Eq.
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#3.2.1 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.#3.2.1 we get:x-(1/2)= √ 49/4 Add 1/2 to both sides to obtain:x = một nửa + √ 49/4 Since a square root has two values, one positive và the other negativex2 - x - 12 = 0has two solutions:x = một nửa + √ 49/4 orx = 50% - √ 49/4 chú ý that √ 49/4 can be written as√49 / √4which is 7 / 2
Solve Quadratic Equation using the Quadratic Formula
3.3Solvingx2-x-12 = 0 by the Quadratic Formula.According lớn 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= 1B= -1C=-12 Accordingly,B2-4AC=1 - (-48) = 49Applying the quadratic formula : 1 ± √ 49 x=—————2Can √ 49 be simplified ?Yes!The prime factorization of 49is7•7 to lớn be able khổng lồ remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Second root).√ 49 =√7•7 =±7 •√ 1 =±7 So now we are looking at:x=(1±7)/2Two real solutions:x =(1+√49)/2=(1+7)/2= 4.000 or:x =(1-√49)/2=(1-7)/2= -3.000