![]() ![]() To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Īccording to the Quadratic Formula, k, the solution for Ak 2+Bk+C = 0, where A, B and C are numbers, often called coefficients, is given by : ![]() Solve Quadratic Equation using the Quadratic FormulaĤ.3 Solving 10k 2+9k-1 = 0 by the Quadratic Formula. Since a square root has two values, one positive and the other negative Now, applying the Square Root Principle to Eq. #4.2.1 we get: The Square Root Principle says that When two things are equal, their square roots are equal. Then, according to the law of transitivity, Things which are equal to the same thing are also equal to one another. Now the clever bit: Take the coefficient of k , which is 9/10 , divide by two, giving 9/20 , and finally square it giving 81/400Īdd 81/400 to both sides of the equation :ġ/10 + 81/400 The common denominator of the two fractions is 400 Adding (40/400)+(81/400) gives 121/400Īdding 81/400 has completed the left hand side into a perfect square : ĭivide both sides of the equation by 10 to have 1 as the coefficient of the first term : Solve Quadratic Equation by Completing The SquareĤ.2 Solving 10k 2+9k-1 = 0 by Completing The Square. Or y = -3.025 Parabola, Graphing Vertex and X-Intercepts : Plugging into the parabola formula -0.4500 for k we can calculate the y -coordinate : For this reason we want to be able to find the coordinates of the vertex.įor any parabola, Ak 2+Bk+C,the k -coordinate of the vertex is given by -B/(2A) . The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. That is, if the parabola has indeed two real solutions. 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. We know this even before plotting "y" because the coefficient of the first term, 10 , is positive (greater than zero).Įach parabola has a vertical line of symmetry that passes through its vertex. Our parabola opens up and accordingly has a lowest point (AKA absolute minimum). Parabolas have a highest or a lowest point called the Vertex. let us now solve the equation by Completing The Square and by using the Quadratic Formula Parabola, Finding the Vertex : ![]() Supplement : Solving Quadratic Equation Directly Solving 10k 2+9k-1 = 0 directlyĮarlier we factored this polynomial by splitting the middle term. Subtract 1 from both sides of the equation : ![]() In other words, we are going to solve as many equations as there are terms in the productĪny solution of term = 0 solves product = 0 as well. We shall now solve each term = 0 separately When a product of two or more terms equals zero, then at least one of the terms must be zero. Which is the desired factorization Equation at the end of step 2 : (10k - 1) ģ.1 A product of several terms equals zero. Step-5 : Add up the four terms of step 4 : Step-4 : Add up the first 2 terms, pulling out like factors :Īdd up the last 2 terms, pulling out common factors : Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 10 Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is 9. Step-1 : Multiply the coefficient of the first term by the constant 10 The middle term is, +9k its coefficient is 9. The first term is, 10k 2 its coefficient is 10. Step 2 : Trying to factor by splitting the middle term Step by step solution : Step 1 : Equation at the end of step 1 : ((2 ![]()
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