5/3x^2-x=1/3
This solution đơn hàng with adding, subtracting & finding the least common multiple.
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Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is khổng lồ the right of the equal sign from both sides of the equation : 5/3*x^2-x-(1/3)=0Step by step solution :
Step 1 :
1 Simplify — 3Equation at the end of step 1 : 5 1 ((— • (x2)) - x) - — = 0 3 3
Step 2 :
5 Simplify — 3Equation at the kết thúc of step 2 : 5 1 ((— • x2) - x) - — = 0 3 3Step 3 :
Equation at the over of step 3 : 5x2 1 (——— - x) - — = 0 3 3Step 4 :
Rewriting the whole as an Equivalent Fraction :4.1Subtracting a whole from a fraction Rewrite the whole as a fraction using 3 as the denominator :x x • 3 x = — = ————— 1 3 Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction và the other fraction involved in the calculation mô tả the same denominator
Adding fractions that have a common denominator :4.2 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 lowest terms if possible:
5x2 - (x • 3) 5x2 - 3x ————————————— = ———————— 3 3 Equation at the end of step 4 : (5x2 - 3x) 1 —————————— - — = 0 3 3
Step 5 :
Step 6 :
Pulling out like terms :6.1 Pull out lượt thích factors:5x2 - 3x=x•(5x - 3)Adding fractions which have a common denominator :6.2 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:
x • (5x-3) - (1) 5x2 - 3x - 1 ———————————————— = ———————————— 3 3 Trying lớn factor by splitting the middle term6.3Factoring 5x2 - 3x - 1 The first term is, 5x2 its coefficient is 5.The middle term is, -3x its coefficient is -3.The last term, "the constant", is -1Step-1 : Multiply the coefficient of the first term by the constant 5•-1=-5Step-2 : Find two factors of -5 whose sum equals the coefficient of the middle term, which is -3.
| -5 | + | 1 | = | -4 | ||
| -1 | + | 5 | = | 4 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored
Equation at the over of step 6 : 5x2 - 3x - 1 ———————————— = 0 3
Step 7 :
When a fraction equals zero :7.1 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:5x2-3x-1 ———————— • 3 = 0 • 3 3 Now, on the left hand side, the 3 cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:5x2-3x-1=0
Parabola, Finding the Vertex:7.2Find the Vertex ofy = 5x2-3x-1Parabolas 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,5, 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 lớn be able lớn 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 0.3000Plugging into the parabola formula 0.3000 for x we can calculate the y-coordinate:y = 5.0 * 0.30 * 0.30 - 3.0 * 0.30 - 1.0 or y = -1.450
Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 5x2-3x-1 Axis of Symmetry (dashed) x= 0.30 Vertex at x,y = 0.30,-1.45 x-Intercepts (Roots) : Root 1 at x,y = -0.24, 0.00 Root 2 at x,y = 0.84, 0.00
Solve Quadratic Equation by Completing The Square7.3Solving5x2-3x-1 = 0 by Completing The Square.Divide both sides of the equation by 5 khổng lồ have 1 as the coefficient of the first term :x2-(3/5)x-(1/5) = 0Add 1 tháng 5 to both side of the equation : x2-(3/5)x = 1/5Now the clever bit: Take the coefficient of x, which is 3/5, divide by two, giving 3/10, & finally square it giving 9/100Add 9/100 lớn both sides of the equation :On the right hand side we have:1/5+9/100The common denominator of the two fractions is 100Adding (20/100)+(9/100) gives 29/100So adding khổng lồ both sides we finally get:x2-(3/5)x+(9/100) = 29/100Adding 9/100 has completed the left hand side into a perfect square :x2-(3/5)x+(9/100)=(x-(3/10))•(x-(3/10))=(x-(3/10))2 Things which are equal to the same thing are also equal to lớn one another.
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Sincex2-(3/5)x+(9/100) = 29/100 andx2-(3/5)x+(9/100) = (x-(3/10))2 then, according to lớn the law of transitivity,(x-(3/10))2 = 29/100We"ll refer lớn this Equation as Eq. #7.3.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(3/10))2 is(x-(3/10))2/2=(x-(3/10))1=x-(3/10)Now, applying the Square Root Principle khổng lồ Eq.#7.3.1 we get:x-(3/10)= √ 29/100 add 3/10 lớn both sides to lớn obtain:x = 3/10 + √ 29/100 Since a square root has two values, one positive & the other negativex2 - (3/5)x - (1/5) = 0has two solutions:x = 3/10 + √ 29/100 orx = 3/10 - √ 29/100 cảnh báo that √ 29/100 can be written as√29 / √100which is √29 / 10