Reformatting the input :Changes do to your input should not impact the solution: (1): "x2" was replaced by "x^2".
You are watching: Solve x2 – 14x + 31 = 63 for x.
Rearrange:Rearrange the equation by subtracting what is come the best of the equal sign from both political parties of the equation : x^2-14*x+31-(63)=0
Step by step solution :
Step 1 :Trying to element by dividing the middle term
1.1Factoring x2-14x-32 The very first term is, x2 its coefficient is 1.The center term is, -14x the coefficient is -14.The critical term, "the constant", is -32Step-1 : main point the coefficient of the an initial term by the continuous 1•-32=-32Step-2 : find two determinants of -32 who sum equates to the coefficient of the middle term, i m sorry is -14.
Step-3 : Rewrite the polynomial splitting the middle term making use of the two components found in step2above, -16 and 2x2 - 16x+2x - 32Step-4 : include up the an initial 2 terms, pulling out favor factors:x•(x-16) include up the critical 2 terms, pulling out typical factors:2•(x-16) Step-5:Add increase the 4 terms that step4:(x+2)•(x-16)Which is the desired factorizationEquation in ~ the end of action 1 :
(x + 2) • (x - 16) = 0
Step 2 :Theory - root of a product :2.1 A product of numerous terms amounts to zero.When a product of two or an ext terms amounts to zero, then at least one of the terms have to be zero.We shall now solve each term = 0 separatelyIn other words, we room going to deal with as many equations as there room terms in the productAny solution of ax = 0 solves product = 0 as well.
Solving a solitary Variable Equation:2.2Solve:x+2 = 0Subtract 2 from both political parties of the equation:x = -2
Solving a solitary Variable Equation:2.3Solve:x-16 = 0Add 16 to both sides of the equation:x = 16
Supplement : fixing Quadratic Equation DirectlySolving x2-14x-32 = 0 directly Earlier we factored this polynomial by splitting the middle term. Allow us now solve the equation by completing The Square and also by utilizing the Quadratic Formula
Parabola, detect the Vertex:3.1Find the crest ofy = x2-14x-32Parabolas have a highest possible or a lowest allude called the Vertex.Our parabola opens up up and accordingly has a lowest allude (AKA pure minimum).We recognize this even prior to plotting "y" since the coefficient the the very first term,1, is confident (greater 보다 zero).Each parabola has actually a vertical line of symmetry the passes through its vertex. As such symmetry, the line of the opposite would, because that example, pass through the midpoint the the two x-intercepts (roots or solutions) of the parabola. The is, if the parabola has indeed two real solutions.Parabolas deserve to model countless real life situations, such as the height over ground, of an object thrown upward, after ~ some duration of time. The peak of the parabola can administer us v information, such together the maximum height that object, thrown upwards, deserve to reach. Therefore we desire to have the ability to find the collaborates of the vertex.For any type of parabola,Ax2+Bx+C,the x-coordinate that the peak is offered by -B/(2A). In our situation the x name: coordinates is 7.0000Plugging right into the parabola formula 7.0000 because that x we can calculate the y-coordinate:y = 1.0 * 7.00 * 7.00 - 14.0 * 7.00 - 32.0 or y = -81.000Parabola, Graphing Vertex and also X-Intercepts :
Root plot because that : y = x2-14x-32 Axis of symmetry (dashed) x= 7.00 Vertex at x,y = 7.00,-81.00 x-Intercepts (Roots) : root 1 in ~ x,y = -2.00, 0.00 source 2 in ~ x,y = 16.00, 0.00Solve Quadratic Equation by completing The Square
3.2Solvingx2-14x-32 = 0 by completing The Square.Add 32 to both next of the equation : x2-14x = 32Now the clever bit: take the coefficient of x, i m sorry is 14, divide by two, providing 7, and also finally square it providing 49Add 49 to both political parties of the equation :On the appropriate hand side we have:32+49or, (32/1)+(49/1)The typical denominator the the two fractions is 1Adding (32/1)+(49/1) provides 81/1So including to both political parties we ultimately get:x2-14x+49 = 81Adding 49 has actually completed the left hand side into a perfect square :x2-14x+49=(x-7)•(x-7)=(x-7)2 things which space equal come the same thing are likewise equal to one another. Sincex2-14x+49 = 81 andx2-14x+49 = (x-7)2 then, according to the regulation of transitivity,(x-7)2 = 81We"ll refer to this Equation together Eq.
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#3.2.1 The Square root Principle claims that when two things room equal, their square roots are equal.Note that the square root of(x-7)2 is(x-7)2/2=(x-7)1=x-7Now, applying the Square source Principle come Eq.#3.2.1 we get:x-7= √ 81 include 7 to both political parties to obtain:x = 7 + √ 81 since a square root has actually two values, one positive and also the various other negativex2 - 14x - 32 = 0has two solutions:x = 7 + √ 81 orx = 7 - √ 81
Solve Quadratic Equation making use of the Quadratic Formula
3.3Solvingx2-14x-32 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the systems forAx2+Bx+C= 0 , where A, B and also C are numbers, often called coefficients, is offered by :-B± √B2-4ACx = ————————2A In our case,A= 1B=-14C=-32 Accordingly,B2-4AC=196 - (-128) = 324Applying the quadratic formula : 14 ± √ 324 x=——————2Can √ 324 be simplified ?Yes!The element factorization that 324is2•2•3•3•3•3 To have the ability to remove something native under the radical, there need to be 2 instances of it (because we space taking a square i.e. 2nd root).√ 324 =√2•2•3•3•3•3 =2•3•3•√ 1 =±18 •√ 1 =±18 So now we room looking at:x=(14±18)/2Two genuine solutions:x =(14+√324)/2=7+9= 16.000 or:x =(14-√324)/2=7-9= -2.000