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(1): "x2" was replaced by "x^2". Step by step solution :Step 1 :Trying to factor by splitting the middle term1.1 Factoring x2-11x-4 The first
term is, x2 its coefficient is 1 . Step-1 : Multiply the coefficient of the first term by the constant 1 • -4 = -4 Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is -11 .
Equation at the end of step 1 : x2 - 11x - 4 = 0
Step 2 :Parabola, Finding the Vertex : 2.1 Find the Vertex of y = x2-11x-4Parabolas have a highest or a
lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , 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 model 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 to 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 5.5000 Plugging into the parabola formula 5.5000 for x we can calculate the y -coordinate : Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = x2-11x-4 Solve Quadratic Equation by Completing The Square 2.2 Solving x2-11x-4 = 0 by Completing The Square .Add 4 to both side of the equation : Now the clever bit: Take the coefficient of x , which is 11 , divide by two, giving 11/2 , and finally square it giving 121/4 Add 121/4 to both sides of the equation : Adding 121/4 has completed the left hand side into a perfect square : We'll refer to this Equation as Eq. #2.2.1 The Square Root Principle says that When two things are equal, their square roots are equal. Note that the square root of Now, applying the Square Root Principle to Eq. #2.2.1 we get: Add 11/2 to both sides to obtain: Since a square root has two values, one positive and the other negative Note that √ 137/4 can be written as Solve Quadratic Equation using the Quadratic Formula 2.3 Solving x2-11x-4 = 0 by the Quadratic
Formula .According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by : 11
± √ 137 Two real solutions: x =(11+√137)/2=11.352 or: x =(11-√137)/2=-0.352 Two solutions were found :
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