Solve quadratic equations 6x



Rearrange the equation by subtracting what is lớn the right of the equal sign from both sides of the equation : 6*x-1-(2/x)=0

Step by step solution :

Step 1 :

2 Simplify — xEquation at the over of step 1 : 2 (6x - 1) - — = 0 x

Step 2 :

Rewriting the whole as an Equivalent Fraction :2.1Subtracting a fraction from a whole Rewrite the whole as a fraction using x as the denominator :

6x - 1 (6x - 1) • x 6x - 1 = —————— = ———————————— 1 x Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction & the other fraction involved in the calculation cốt truyện the same denominator

Adding fractions that have a common denominator :2.2 Adding up the two equivalent fractions add 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 lowest terms if possible:

(6x-1) • x - (2) 6x2 - x - 2 ———————————————— = ——————————— x x Trying khổng lồ factor by splitting the middle term2.3Factoring 6x2 - x - 2 The first term is, 6x2 its coefficient is 6.The middle term is, -x its coefficient is -1.The last term, "the constant", is -2Step-1 : Multiply the coefficient of the first term by the constant 6•-2=-12Step-2 : Find two factors of -12 whose sum equals the coefficient of the middle term, which is -1.

-4+3=-1That"s it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 và 36x2 - 4x+3x - 2Step-4 : địa chỉ up the first 2 terms, pulling out like factors:2x•(3x-2) showroom up the last 2 terms, pulling out common factors:1•(3x-2) Step-5:Add up the four terms of step4:(2x+1)•(3x-2)Which is the desired factorization

Equation at the end of step 2 : (3x - 2) • (2x + 1) ——————————————————— = 0 x

Step 3 :

When a fraction equals zero :3.1 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:

(3x-2)•(2x+1) ————————————— • x = 0 • x x Now, on the left hand side, the x cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:(3x-2) • (2x+1)=0

Theory - Roots of a product :3.2 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 khổng lồ 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:3.3Solve:3x-2 = 0Add 2 khổng lồ both sides of the equation:3x = 2 Divide both sides of the equation by 3:x = 2/3 = 0.667

Solving a Single Variable Equation:3.4Solve:2x+1 = 0Subtract 1 from both sides of the equation:2x = -1 Divide both sides of the equation by 2:x = -1/2 = -0.500

Supplement : Solving Quadratic Equation Directly

Solving 6x2-x-2 = 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 = 6x2-x-2Parabolas 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,6, 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.

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For this reason we want 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 0.0833Plugging into the parabola formula 0.0833 for x we can calculate the y-coordinate:y = 6.0 * 0.08 * 0.08 - 1.0 * 0.08 - 2.0 or y = -2.042

Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 6x2-x-2 Axis of Symmetry (dashed) x= 0.08 Vertex at x,y = 0.08,-2.04 x-Intercepts (Roots) : Root 1 at x,y = -0.50, 0.00 Root 2 at x,y = 0.67, 0.00

Solve Quadratic Equation by Completing The Square

4.2Solving6x2-x-2 = 0 by Completing The Square.Divide both sides of the equation by 6 to lớn have 1 as the coefficient of the first term :x2-(1/6)x-(1/3) = 0Add 1/3 to both side of the equation : x2-(1/6)x = 1/3Now the clever bit: Take the coefficient of x, which is 1/6, divide by two, giving 1/12, and finally square it giving 1/144Add 1/144 to lớn both sides of the equation :On the right hand side we have:1/3+1/144The common denominator of the two fractions is 144Adding (48/144)+(1/144) gives 49/144So adding khổng lồ both sides we finally get:x2-(1/6)x+(1/144) = 49/144Adding 1/144 has completed the left hand side into a perfect square :x2-(1/6)x+(1/144)=(x-(1/12))•(x-(1/12))=(x-(1/12))2 Things which are equal to the same thing are also equal lớn one another. Sincex2-(1/6)x+(1/144) = 49/144 andx2-(1/6)x+(1/144) = (x-(1/12))2 then, according to lớn the law of transitivity,(x-(1/12))2 = 49/144We"ll refer khổng lồ this Equation as Eq. #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-(1/12))2 is(x-(1/12))2/2=(x-(1/12))1=x-(1/12)Now, applying the Square Root Principle lớn Eq.#4.2.1 we get:x-(1/12)= √ 49/144 địa chỉ 1/12 lớn both sides lớn obtain:x = 1/12 + √ 49/144 Since a square root has two values, one positive and the other negativex2 - (1/6)x - (1/3) = 0has two solutions:x = 1/12 + √ 49/144 orx = 1/12 - √ 49/144 note that √ 49/144 can be written as√49 / √144which is 7 / 12

Solve Quadratic Equation using the Quadratic Formula

4.3Solving6x2-x-2 = 0 by the Quadratic Formula.According khổng lồ 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= 6B= -1C= -2 Accordingly,B2-4AC=1 - (-48) = 49Applying the quadratic formula : 1 ± √ 49 x=—————12Can √ 49 be simplified ?Yes!The prime factorization of 49is7•7 khổng lồ be able to remove something from under the radical, there have to lớn 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)/12Two real solutions:x =(1+√49)/12=(1+7)/12= 0.667 or:x =(1-√49)/12=(1-7)/12= -0.500