3 x + 5 x = 6x + 2

     

Step by step solution :

Step 1 :

Equation at the kết thúc of step 1 : (5x2 + 6x) - 5 = 0

Step 2 :

Trying lớn factor by splitting the middle term2.1Factoring 5x2+6x-5 The first term is, 5x2 its coefficient is 5.The middle term is, +6x its coefficient is 6.The last term, "the constant", is -5Step-1 : Multiply the coefficient of the first term by the constant 5•-5=-25Step-2 : Find two factors of -25 whose sum equals the coefficient of the middle term, which is 6.

-25+1=-24
-5+5=0
-1+25=24

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Equation at the kết thúc of step 2 :

5x2 + 6x - 5 = 0

Step 3 :

Parabola, Finding the Vertex:3.1Find the Vertex ofy = 5x2+6x-5Parabolas 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,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.


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For this reason we want khổng lồ be able to 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 -0.6000Plugging into the parabola formula -0.6000 for x we can calculate the y-coordinate:y = 5.0 * -0.60 * -0.60 + 6.0 * -0.60 - 5.0 or y = -6.800

Parabola, Graphing Vertex và X-Intercepts :

Root plot for : y = 5x2+6x-5 Axis of Symmetry (dashed) x=-0.60 Vertex at x,y = -0.60,-6.80 x-Intercepts (Roots) : Root 1 at x,y = -1.77, 0.00 Root 2 at x,y = 0.57, 0.00

Solve Quadratic Equation by Completing The Square

3.2Solving5x2+6x-5 = 0 by Completing The Square.Divide both sides of the equation by 5 to lớn have 1 as the coefficient of the first term :x2+(6/5)x-1 = 0Add 1 lớn both side of the equation : x2+(6/5)x = 1Now the clever bit: Take the coefficient of x, which is 6/5, divide by two, giving 3/5, & finally square it giving 9/25Add 9/25 lớn both sides of the equation :On the right hand side we have:1+9/25or, (1/1)+(9/25)The common denominator of the two fractions is 25Adding (25/25)+(9/25) gives 34/25So adding to lớn both sides we finally get:x2+(6/5)x+(9/25) = 34/25Adding 9/25 has completed the left hand side into a perfect square :x2+(6/5)x+(9/25)=(x+(3/5))•(x+(3/5))=(x+(3/5))2 Things which are equal to the same thing are also equal to lớn one another. Sincex2+(6/5)x+(9/25) = 34/25 andx2+(6/5)x+(9/25) = (x+(3/5))2 then, according to lớn the law of transitivity,(x+(3/5))2 = 34/25We"ll refer to lớn this Equation as Eq. #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+(3/5))2 is(x+(3/5))2/2=(x+(3/5))1=x+(3/5)Now, applying the Square Root Principle lớn Eq.#3.2.1 we get:x+(3/5)= √ 34/25 Subtract 3/5 from both sides to obtain:x = -3/5 + √ 34/25 Since a square root has two values, one positive and the other negativex2 + (6/5)x - 1 = 0has two solutions:x = -3/5 + √ 34/25 orx = -3/5 - √ 34/25 note that √ 34/25 can be written as√34 / √25which is √34 / 5

Solve Quadratic Equation using the Quadratic Formula

3.3Solving5x2+6x-5 = 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= 5B= 6C= -5 Accordingly,B2-4AC=36 - (-100) = 136Applying the quadratic formula : -6 ± √ 136 x=——————10Can √ 136 be simplified ?Yes!The prime factorization of 136is2•2•2•17 khổng lồ be able lớn remove something from under the radical, there have lớn be 2 instances of it (because we are taking a square i.e. Second root).√ 136 =√2•2•2•17 =±2 •√ 34 √ 34 , rounded to lớn 4 decimal digits, is 5.8310So now we are looking at:x=(-6±2• 5.831 )/10Two real solutions:x =(-6+√136)/10=(-3+√ 34 )/5= 0.566 or:x =(-6-√136)/10=(-3-√ 34 )/5= -1.766

Two solutions were found :

x =(-6-√136)/10=(-3-√ 34 )/5= -1.766 x =(-6+√136)/10=(-3+√ 34 )/5= 0.566