Reformatting the entry :Changes make to your input need to not affect the solution: (1): "r2" was replaced by "r^2".
You are watching: Find the possible value or values of r in the quadratic equation r2 – 7r – 8 = 0.
Step by step solution :
Step 1 :Trying to aspect by dividing the center term
1.1Factoring r2-7r-8 The first term is, r2 its coefficient is 1.The middle term is, -7r that is coefficient is -7.The last term, "the constant", is -8Step-1 : main point the coefficient of the an initial term through the constant 1•-8=-8Step-2 : find two components of -8 whose sum equals the coefficient that the middle term, i beg your pardon is -7.
Step-3 : Rewrite the polynomial separating the middle term using the two factors found in step2above, -8 and also 1r2 - 8r+1r - 8Step-4 : include up the an initial 2 terms, pulling out favor factors:r•(r-8) add up the critical 2 terms, pulling out usual factors:1•(r-8) Step-5:Add up the 4 terms of step4:(r+1)•(r-8)Which is the desired factorizationEquation in ~ the end of action 1 :
(r + 1) • (r - 8) = 0
Step 2 :Theory - root of a product :2.1 A product of number of terms equals zero.When a product of two or an ext terms equates to zero, climate at least one of the terms must be zero.We shall currently solve each term = 0 separatelyIn various other words, we space going to solve as many equations as there space terms in the productAny systems of ax = 0 solves product = 0 together well.
Solving a solitary Variable Equation:2.2Solve:r+1 = 0Subtract 1 from both political parties of the equation:r = -1
Solving a single Variable Equation:2.3Solve:r-8 = 0Add 8 to both political parties of the equation:r = 8
Supplement : resolving Quadratic Equation DirectlySolving r2-7r-8 = 0 straight Earlier we factored this polynomial by dividing the center term. Let us currently solve the equation by completing The Square and also by using the Quadratic Formula
Parabola, finding the Vertex:3.1Find the vertex ofy = r2-7r-8Parabolas have a greatest or a lowest allude called the Vertex.Our parabola opens up up and accordingly has a lowest suggest (AKA absolute minimum).We know this even prior to plotting "y" because the coefficient that the very first term,1, is confident (greater than zero).Each parabola has actually a vertical heat of symmetry the passes through its vertex. As such symmetry, the line of symmetry would, because that example, pass with the midpoint of the 2 x-intercepts (roots or solutions) the the parabola. The is, if the parabola has indeed two actual solutions.Parabolas can model countless real life situations, such together the height above ground, of an item thrown upward, after ~ some duration of time. The crest of the parabola can administer us through information, such together the maximum elevation that object, thrown upwards, have the right to reach. For this reason we desire to have the ability to find the collaborates of the vertex.For any type of parabola,Ar2+Br+C,the r-coordinate of the peak is offered by -B/(2A). In our instance the r coordinate is 3.5000Plugging into the parabola formula 3.5000 because that r we have the right to calculate the y-coordinate:y = 1.0 * 3.50 * 3.50 - 7.0 * 3.50 - 8.0 or y = -20.250Parabola, Graphing Vertex and also X-Intercepts :
Root plot for : y = r2-7r-8 Axis of the opposite (dashed) r= 3.50 Vertex at r,y = 3.50,-20.25 r-Intercepts (Roots) : source 1 in ~ r,y = -1.00, 0.00 source 2 at r,y = 8.00, 0.00Solve Quadratic Equation by completing The Square
3.2Solvingr2-7r-8 = 0 by completing The Square.Add 8 come both side of the equation : r2-7r = 8Now the clever bit: take it the coefficient of r, i beg your pardon is 7, divide by two, offering 7/2, and also finally square it providing 49/4Add 49/4 to both political parties of the equation :On the ideal hand side we have:8+49/4or, (8/1)+(49/4)The typical denominator the the two fractions is 4Adding (32/4)+(49/4) gives 81/4So including to both sides we finally get:r2-7r+(49/4) = 81/4Adding 49/4 has completed the left hand side right into a perfect square :r2-7r+(49/4)=(r-(7/2))•(r-(7/2))=(r-(7/2))2 things which are equal to the exact same thing are also equal to one another. Sincer2-7r+(49/4) = 81/4 andr2-7r+(49/4) = (r-(7/2))2 then, follow to the legislation of transitivity,(r-(7/2))2 = 81/4We"ll describe this Equation together Eq.
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#3.2.1 The Square source Principle says that once two things room equal, their square roots are equal.Note the the square source of(r-(7/2))2 is(r-(7/2))2/2=(r-(7/2))1=r-(7/2)Now, applying the Square root Principle come Eq.#3.2.1 us get:r-(7/2)= √ 81/4 add 7/2 to both sides to obtain:r = 7/2 + √ 81/4 due to the fact that a square root has actually two values, one positive and also the various other negativer2 - 7r - 8 = 0has 2 solutions:r = 7/2 + √ 81/4 orr = 7/2 - √ 81/4 note that √ 81/4 deserve to be created as√81 / √4which is 9 / 2
Solve Quadratic Equation using the Quadratic Formula
3.3Solvingr2-7r-8 = 0 by the Quadratic Formula.According to the Quadratic Formula,r, the systems forAr2+Br+C= 0 , whereby A, B and C space numbers, often referred to as coefficients, is provided by :-B± √B2-4ACr = ————————2A In our case,A= 1B= -7C= -8 Accordingly,B2-4AC=49 - (-32) = 81Applying the quadratic formula : 7 ± √ 81 r=—————2Can √ 81 be simplified ?Yes!The prime factorization of 81is3•3•3•3 To be able to remove something indigenous under the radical, there have to be 2 instances of the (because we space taking a square i.e. 2nd root).√ 81 =√3•3•3•3 =3•3•√ 1 =±9 •√ 1 =±9 So currently we space looking at:r=(7±9)/2Two genuine solutions:r =(7+√81)/2=(7+9)/2= 8.000 or:r =(7-√81)/2=(7-9)/2= -1.000