Reformatting the input :Changes made to your input should not affect the solution: Show (1): "x2" was replaced by "x^2". Step by step solution :Step 1 :Equation at the end of step 1 : (3x2 - 14x) - 16 = 0
Step 2 :Trying to factor by splitting the middle term2.1 Factoring 3x2-14x-16 The first term is, 3x2 its coefficient is 3 . Step-1 : Multiply the coefficient of the first term by the constant 3 • -16 = -48 Step-2 : Find two factors of -48 whose sum equals the coefficient of the middle term, which is -14 .
Equation at the end of step 2 : 3x2 - 14x - 16 = 0
Step 3 :Parabola, Finding the Vertex : 3.1 Find the Vertex of y = 3x2-14x-16Parabolas 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, 3 , 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 2.3333 Plugging into the parabola formula 2.3333 for x we can calculate the y -coordinate : Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 3x2-14x-16 Solve Quadratic Equation by Completing The Square 3.2 Solving 3x2-14x-16 = 0 by Completing The Square .Divide both sides of the equation by 3 to have 1 as the coefficient of the first term : Add 16/3 to both side of the equation : Now the clever bit: Take the coefficient of x , which is 14/3 , divide by two, giving 7/3 , and finally square it giving 49/9 Add 49/9 to both sides
of the equation : Adding 49/9
has completed the left hand side into a perfect square : We'll refer to 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 Now, applying the Square Root Principle to Eq. #3.2.1 we get: Add 7/3 to both sides to obtain: Since a square root has two values, one positive and the other negative Note that √ 97/9 can be written as Solve Quadratic Equation using the Quadratic Formula 3.3 Solving 3x2-14x-16 = 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 : 14 ± √ 388 Yes! The prime factorization of 388 is √ 388 = √ 2•2•97
= √ 97 , rounded to 4 decimal digits, is 9.8489 Two real solutions: x =(14+√388)/6=(7+√ 97 )/3= 5.616 or: x =(14-√388)/6=(7-√ 97 )/3= -0.950 Two solutions were found :
What are the solution to the quadratic equation 5y 6 2 24?Answer and Explanation: Thus, the possible solutions are −0.22 and −2.18. − 2.18 .
What are the solutions to the equation x2 6x 40?Summary: The solutions to the equation x2 + 6x = 40 are -10, 4.
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