Properties of a quadratic equation
In the quadratic equation form ax2+bx+c=0, x stands for the unknown value and a, b and c the numbers that are known such that a will never be equal to zero. If in case a is zero, then that equation is not a quadratic equation but a linear equation. a, b and c are the coefficients of the equation which can be distinguished by referring to them as the constant, the linear or the quadratic coefficient or the free term. A quadratic equation can be solved using methods like factoring by the quadratic formula, by graphing or by completing the square. To solve a quadratic equation in one variable can be done through the factoring technique together with the zero factor property as below:
- Begin by writing the quadratic equation in a standard form.
- Factor the quadratic polynomial to a product of linear factors.
- Use Zero Factor Property to make each factor equal to zero.
- Find the solution to each of the resulting linear equations. The resulting solution is the solution of the original quadratic equation.
The quadratic equation is also known as a univariate because it involves only one unknown. The quadratic equation only has powers of x which are non-negative integers and this makes it a polynomial equation and in particular, it is a second-degree polynomial equation because its greatest power is two.
Solving quadratic equations
A quadratic equation which has real or complex coefficients has two solutions known as roots. These solutions can or cannot be distinct and can also be or not be real. The first method of solving a quadratic equation is through factoring by inspection. It is possible to write the quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. And with simple inspection, it is also possible to know the values of p, q, r and s which make the two forms equivalent to each other. If you write the quadratic equation in a second form, the zero factor property states that a quadratic equation is satisfied if px + q = 0 or rx + s = 0. The solution of these two linear equations gives you the roots of the quadratic. Factoring by inspection method is the first method of solving a quadratic equation that students learn. If you are given a quadratic equation in the form x2 + bx + c = 0, the factorization will have the form(x + q)(x + s), and you will have to find q and s that add up to b and whose product is c. For example, x2 + 5x + 6 factors as (x + 3)(x + 2). In the case where a is not equal to 1, you will need more guesses in order to find the solution if at all it can be factored by inspection. Except for special cases like where b = 0 or c = 0, factoring by inspection works for a quadratic equation that has rational roots. This means that most quadratic equations in practical cases cannot be computed through factoring by inspection.
The second method of solving a quadratic equation is by completing the square. The completing by square method is used to come up with a new formula for solving a quadratic equation. The formula is known as the quadratic formula. The quadratic formula’s mathematical proof is.
One property of this equation form is that it gives one valid root when a = 0, while the other root has division by zero, since when a = 0, the quadratic equation changes to a linear equation which has one root. By contrast, the formula contains division by zero in both cases. With the polynomial expansion, you can easily tell that this equation is equivalent to the quadratic equation.
The third method of solving a quadratic equation is by graphing. The graph of a quadratic equation is a parabola that opens up when the leading coefficient is positive and it opens down when the leading is a negative. The x intercepts are crucial and they should be gotten, plotted and labeled. In any function, the x intercepts are gotten by getting the real zeros of that function and the zeros of any given function can be gotten by solving that equation that results from f(x)=0. In a quadratic function, the function f of the equation that results from f(x) =0 can be solved by factoring along with the zero factor property or with the quadratic formula. The vertex of a parabola is also crucial and can be gotten, plotted and labeled. In the case of a quadratic function f, the equation resulting from f(x) = 0 is always solvable with the quadratic formula or by factoring in along with the Zero Factor Property.
Graphing the quadratic equations
The first step in graphing a quadratic equation is determining the form of the equation given. A quadratic equation can be expressed in three forms. These forms are the quadratic form, vertex form and standard form. Any of the forms can be used in graphing but the process of graphing each of them is a bit different. The standard form is the form where the equation is written as f(x) = ax2+bx+c whereby a, b and c are real numbers and a is not a zero. Examples of standard form equations are f(x) = x2 +2x+1 and f(x) =9 x2+10x-8. A vertex form is the form whereby the equation is expressed as f(x) =a(x-h) 2 +k whereby a, h and k are real numbers and a is not zero. This equation form is known as a vertex form because h and k will directly give you a central point of the parabola at point (h, k). Examples of vertex form equations are -3(x-5) 2+1 and 9(x-4) 2+18. To graph any of this quadratic equation forms you will first have to get the vertex of the parabola which is the middle point (h, k) at the tip of the curve. The vertex coordinates in the standard form are given by k=f(h) and h=-b/2a. In the vertex equation form, h and k are gotten directly from the equation.
The second step in graphing a quadratic equation is to define the variables. To solve a quadratic equation, the variables a, b and c or a, h and k must be defined. Common algebra problems will give you a quadratic equation with variables already filled in a vertex or standard form. An example of a standard equation form with variables is f(x)= 2×2+16x+39, so a is 2, b is 16 and c is 39. An example of a vertex equation form with variables is f(x)=4(x-5) 2+12, a is 4, b is 16 and k is 12. The third step is to find h. In the vertex form, h is already provided. In standard form h is calculated by h =-b/2a and so in f(x)= 2×2+16x+39 h will be -16/2(2). After solving you will get h=-4. In the vertex form f(x)=4(x-5) 2+12, h is equal to 5.
The third step is to find k. In the vertex form, k is already known just like h but for standard form k is equal to f(h). This means that you can get k in the standard form equation by replacing every x with the value of h. So k=2(-4) 2 +16(-4)+39, k=2(16)-64+39, k=32-64+39 =7. The value of k in the vertex quadratic equation form is 12.
The fifth step is to plot the vertex. The parabola vertex is at point (h, k). The h represents the x coordinate and the k represents the y coordinate. The vertex is the middle point of the parabola. In the standard equation form, the vertex will be at point (-4, 7). This point will be plotted on the graph and labeled. In the vertex equation form, the vertex is at (5, 12).
The sixth step is to draw the parabola axis. The axis of symmetry in a parabola is a line that runs through the middle and divides the parabola into half. Across the axis, the right side of a parabola will mirror the left side of the parabola. For the quadratic equation of the form ax2+bx+c or a(x-h) 2+k, the axis is the line which is vertical and passes through the vertex. In the standard form equation, the axis is the line that is parallel to the y axis and passes through the point (-4, 7). This line is not part of the parabola but it shows you how a parabola curves symmetrically.
The seventh step is to get the direction of opening. After knowing the axis and vertex of the parabola determine if the parabola opens downwards or upwards. If a is positive, the parabola opens upwards but if a is negative the parabola will open downwards. For the standard form equation f(x)= 2×2+16x+39, the parabola will open upwards since a=2(positive), and in the vertex equation form f(x)=4(x-5) 2+12, the parabola will also open upwards since a=4(positive).
The eight step if necessary, you can find and plot the x intercepts. The x intercepts are the two points where the parabola meets the x axis. Not all parabolas have x intercepts. If the parabola is a vertex that opens upwards and has the vertex above the x axis or if it opens downwards and has a vertex below the x axis, it will not have x intercepts. You can also find and plot the y intercept too. To find the y intercept, you will set x to zero and then solve the equation for y or f(x). This will give you the value of y when the parabola passes through the y axis. And unlike the x intercept, standard parabolas can have one y intercept and for the standard quadratic equation forms, the y intercept is at y=c.
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