How to Solve Quadratic Equations
Factoring, Quadratic Formula, & Quadratic Calculator
By P. X. WilliamsIn algebra, a quadratic equation is a second-degree polynomial equation of the form
0 = Ax^{2} + Bx + C.
If A, B, and C are real numbers, then the solution set of a quadratic equation can be either two real values, one real value repeated, or two imaginary values that are complex conjugates of one another. If you graph a parabolic equation of the form y = Ax^{2} + Bx + C, the points where the parabola crosses the x-axis are the solutions. If a parabola does not intersect the x-axis, it means that the roots are either imaginary or complex numbers.
Quadratic equations can be solved by either factoring the polynomial into two linear factors and setting each equal to zero, or by plugging A, B, and C into the Quadratic Formula or a quadratic calculator.
Linear factoring is not possible when the roots are imaginary, complex, or irrational. However, the Quadratic Formula always yields the correct solution. You can also use the quadratic solver above. (Your browser must allow JavaScript, usually the default setting.)
Factoring
The first step in factoring is to divide both sides of the equation by A. For example, if you have 0 = 2x^{2} + 20x + 48, you can divide both sides of the equation by 2 to obtain 0 = x^{2} + 10x + 24.Next, look at the coefficient of the x term, 10, and the constant term, 24. You need to find two numbers that add up to 10 and have a product of 24. The correct pair of numbers is 6 and 4.
Now, factor the polynomial into 0 = (x + 6)(x + 4). To find the solution set for this equation, you set (x + 6) equal to zero, and set (x + 4) equal to zero. This gives you two solutions: x = -6 and x = -4.
Here's another example: 0 = x^{2} - 14x + 49. The only pair of numbers that adds up to -14 and multiplies to 49 is -7 and -7. Thus, the equation factors into 0 = (x - 7)(x - 7). The only solution to the equation is x = 7, since the same factor is repeated twice.
Quadratic Formula
Not all quadratic equations can be solved by factoring. For example, the equation 0 = x^{2} + 9x + 50 cannot be factored. There is no pair of real numbers that adds up to 9 and has a product of 50. The solutions to this equation are imaginary numbers that can only be found with the quadratic formula.The quadratic formula is derived by completing the square. Notice how the general quadratic equation can be solved for x:
0 = Ax^{2} + Bx + C
0 = Ax^{2} + Bx + B^{2}/(4A) - B^{2}/(4A) + C
0 = A[x^{2} + (B/A)x + B^{2}/(4A^{2})] - B^{2}/(4A) + C
0 = A[x + B/(2A)]^{2} - B^{2}/(4A) + C
[B^{2}/(4A) - C]/A = [x + B/(2A)]^{2}
± sqrt[B^{2} - 4AC]/(2A) = x + B/(2A)
-B/(2A) ± sqrt(B^{2} - 4AC)/(2A) = x
To use the quadratic equation to solve x^{2} + 9x + 50, you must set a = 1, b = 9 and c = 50. This gives two answers, x = -4.5 + [sqrt(319)/2]i and x = -4.5 - [sqrt(319)/2]i. The i is the imaginary unit; it is equal to the square root of -1.
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