![]() ![]() Now we can rewrite the original quadratic equation as: Using this formula here with A = x and B = 4, we get: Remember that the formula to factor a difference of squares is: We can see that the left side is a difference of squares: x 2 – 4 2. Example 1: Solving A Quadratic Equation By FactoringĬonsider the following quadratic equation: However, it takes a little practice to get good at spotting when you can factor a quadratic easily. There are lots of cases when it is easier to factor a quadratic equation than to graph, complete the square, or use the quadratic formula. How To Solve Quadratic Equations By Factoring Now we know the solutions to the quadratic equation, along with the standard and factored forms. So the value of a is 3, and the quadratic factored form is: We will use this value in the quadratic to solve for a: Looking at the graph again, we can see that the vertex is indeed at x = 4, with a value of y = -12. This gives us an x-coordinate of x = 4 for the vertex. We know that its x-coordinate will be the average of the zeros x = 2 ad x = 6. To do that, we can use the vertex of the parabola. First, the quadratic factored form looks like this: However, we can find the quadratic equation as well. At this point, we know that the solutions of the equation are x = 2 and x = 6. We can see from the graph that the parabola intersects the x-axis (the line y = 0) at x = 2 and x= 6. We can sometimes find the solutions of a quadratic equation by graphing and finding the x-intercepts (zeros). Example: Solving A Quadratic Equation By GraphingĬonsider the following parabola (the graph of a quadratic): To solve a quadratic equation by graphing, all we really need to do is find out where the zeros are (the points where the graph intersects the x-axis). ![]() How To Solve Quadratic Equations By Graphing Let’s take a look at some examples of each method, starting with graphing. If r and s are real, we also have x-intercepts for the parabola, which makes it easier to graph. If we take the average of the two solutions r and s, we can find the x-coordinate of the parabola’s vertex (that is, the axis of symmetry).If we use the coefficient ‘a’ together with the solutions r and s, we can write the factored form of the quadratic equation as a(x – r)(x – s) = 0.These solutions may be distinct and real (positive discriminant), double real (zero discriminant), or complex conjugates (negative discriminant). When we use the quadratic formula on a quadratic equation, we get two solutions to the equation: r and s.When the method of completing the square is used on the general quadratic equation in standard form, ax 2 + bx + c = 0, we get the quadratic formula as a result.Here is how these methods are connected to one another: The only drawback is that the calculations can become tedious, as they involve multiplication, addition, radicals, and division (fractions). It is derived from using the previous method (complete the square) on a general quadratic in standard form: ax 2 + bx + c = 0. ![]()
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