Quadratic Equation 二次方程

A quadratic equation is one where there is a x² in the equation.  Of course, x is just a representation of an unknown value to be solved.  It can be y or z or p or q.  The following are therefore examples of quadratic equations.

(1) z² – 16z + 64 = 0; (2) y² - 49 = 0; (3) 2w² = 56 – 6w; and (4) x² - 5x = - 6

Quadratic Equation  二次方程  Èr cì fāngchéng

There are four methods of solving a quadratic equation.

The first method is the quadratic formula method. The equation has to be re-arranged into the format: ax² + bx + c = 0. We shall discuss how this can easily be done later. We need to memorise the formula which is:

Let us say we want to solve the quadratic equation x² - 5x = - 6. We need to move all our terms into the right hand side of the equal sign. So the equation becomes x² - 5x + 6 = 0.

So a = + 1; b = - 5; and c = + 6; b² - 4ac = (-5)² - 4 x (+1) x (-5) = 25 - (-20) = 25 + 20 = 45.

Square root of 45 is 6.7 So, x = (- b +/- 6.7) divided by 2a or. x = [-(-5) +/- 6.7] / 2

x = (+ 5 + 6.7) / 2 = 5.85 x = (+ 5 - 6.7) / 2 = - 0.85

The second method is that of factoring. We need to find the two factors of the equation and then with these factors we can have the answers for x. Let us see the video - https://youtu.be/zFSlzjyfc8o. This video shows how the quadratic equation 2x² - 12x + 10 = 0 has been solved.

Step # 1: For all the three terms, we take out the common factor of 2. So 2x² - 12x + 10 = 0 becomes 2(x² - 6x + 5) = 0.

Step # 2: For x² - 6x + 5 = 0, we need to break the second term up into two terms such that when the two coefficients are multiplied the answer is + 5 (Product) and when added together the two coefficients give an answer of - 6. The factors of 5 are 1 and 5. Therefore the second term can be broken up as - x and - 5x (or - 1x and - 5x). So 2(x² - 6x + 5) = 0 becomes 2(x² - x - 5x + 5) = 0.

Step # 3: For x² - x - 5x + 5 = 0, we can take out the common factor x for the first two terms and the common factor 5 for the third and fourth terms. This now becomes: x ( x - 1) - 5 ( x - 1) = 0. This now becomes ( x - 1) (x - 5) = 0.

Step # 4: So either x - 1 = 0 or x - 5 = 0. The solution for this quadratic equation is now either x equals 1 or x equals 5.

The third method of solving a quadratic equation is called ‘Completing the Square’. This is not a common method and can be difficult for some students. In this article therefore, this method will not be explained.

The fourth and last method is that of ‘Graphing’. In this method, we can solve the equation by drawing of a parabola, on a graph with the x-axis (horizontal line) and the y-axis (vertical line). As ‘a’ is positive, the parabola is opening up. We need to find out the value of x, using the formula x = - b / 2a. Watch the video https://youtu.be/pNpTzbZR0JQ for solving the quadratic equation y = 2x² - 4x - 1. So, a = + 2; b = - 4; and c = - 1. So, x = - b / 2a gives us x = 1.

We also need to find the axis of symmetry (AOS) and the vertex, or the lowest point of the parabola. The AOS is the answer we obtained earlier for x which is x = 1. This axis runs parallel to y, one unit to the right of 0. The vertex can be found by substituting x = 1 in the equation y = 2x² - 4x - 1. When x = 1, y = -3. So the vertex is (1, -3). Similarly when x = 0, y = -1. Once we draw this curve on the graph, the answer of x is the two points where the curve meets the x-axis.

Let us try out solving the following quadratic equations, using the quadratic formula:

(1) z² – 16z + 64 = 0; (2) y² - 49 = 0; (3) 2w² = 56 – 6w;