How to Solve a Quadratic Equation

Quadratic equations are those that can be stated as: y = ax^2 + bx + c, where x is the variable, a, b, and c are constants, and the constant a is not 0. Graphically, quadratic equations are described by parabolas. Solving a quadratic equation means finding the "roots" of the equation which are those values of x for which y = 0. Graphically, this means finding those values of x where the parabola intersects the x-axis.There are many methods to solve quadratic equations, such as factoring, completing the square, etc., with each useful for some quadratic equations but not for others. However, the quadratic formula which is explained here will solve any quadratic equation.

Things You'll Need

• Your memory, a piece of paper, and a pen or pencil

Instructions

How to solve quadratic equations

1. Quadratic equations can come in many forms. You may find an equation such as:3 * (x-4) * (x+2) = 0, or (x+1) * (x-1) = 8or x^2 - 4x + 1 = -3.In each case, multiply out any parentheses, and move all terms to the left side until you reach the general form:ax^2 + bx + c =0.In the above examples:3 * (x-4) * (x+2) = 0 when multiplied out becomes 3x^2 - 6x - 24 = 0.(x+1) * (x-1) = 8 when multiplied out becomes x^2 - 1 = 8, and subtracting 8 from both sides becomes x^2 - 9 =0.x^2 - 4x + 1 = -3 when 3 is added to both sides becomes x^2 - 4x + 4 = 0.Note that while a cannot be 0 (or the equation is no longer quadratic), either b or c, or both, can be 0.

2. Identify the values of your parameters, a, b, and c. The parameter a is what multiplies x^2 in your quadratic equation's general form. If you only have x^2, then a = 1. If you have -x^2, then a = -1. If you have, say, 4x^2, then a = 4. The parameter b is what multiplies x in the general form of your quadratic equation. If you have no term with x in it (besides the quadratic x^2) then b = 0. If you have, say -5x in your quadratic equation's general form, then you have b = -5. The free number in the general form of your quadratic equation is the parameter c. If you have no term without x or x^2, the value of c in your quadratic equation is 0.For example, if your quadratic equation's general form is:2x^2 - x - 3 = 0then your parameters are a = 2, b = -1, and c = -3.

3. Write down the following general quadratic formula:x = [-b +/- sqrt(-b - 4ac)]/2a.Then, substitute in the values of a, b, and c you identified in the previous step. In the above example of 2x^2 - x - 3 = 0, the quadratic formula becomes:x = [-(-1) +/- sqrt[-(-1) - 4*2*(-3)]]/(2*2).

4. All that's left is to solve this simple formula. Multiplying out the numbers in the square root we find:x = [1 +/- sqrt(1 + 24)]/4 or:x = [1 +/- sqrt(25)]/4 or:x = (1 +/- 5)/4.Thus, the roots of this quadratic equation are x = 6/4 and x = -4/4, or written as decimal fractions, x = 1.5 and x = -1. Sometimes this is written x = 1.5, -1.

5. There are some quadratic equations where you will find that the number under the square root is negative. In these cases one can say that there are no real roots to the quadratic equation. There are roots, but they are complex numbers (i.e. numbers with imaginary components). In most algebra courses these will not appear. If you do come across such a quadratic equation, simply write "i" in front of the square root, and change the sign of what is inside the square root to a plus. At this point you may not add or subtract the term with the "i" from the term without it. You can, however, divide both by 2a. As an example, if your quadratic equation general form is: x^2 +x + 9.25 = 0 you have a = 1, b = 1, and c = 9.25. Your quadratic formula becomes:x = [-1 +/- sqrt(1 - 4*1*9.25)]/(2*1) = [-1 +/ -sqrt(-36)]/2. This can be rewritten as:x = [-1 +/- i*sqrt(36)]/2 which can be simplified to: x = -0.5 +/- 3i or x = -0.5 + 3i, -0.5 - 3i.

Tips & Warnings

• As mentioned above, quadratic functions are described geometrically by parabolas. If the value of a is positive, the parabola will have a minimum and be open upwards. If a is negative, the parabola is inverted. The parabola then has a maximum, and is open downwards. Since the roots are those values of x where the parabola intersects the x-axis, when there are no real roots, the parabola never intersects the x-axis. If a is positive, the parabola is always above the x-axis. If a is negative, the parabola is always below the x-axis.

• In those cases where the parabola just touches the x-axis, either from above or below, the two roots are equal to each other. This happens when the square root in the quadratic formula becomes zero. These are the only cases where the quadratic equation can be written in the form (x + k)^2 = 0 with k being a real constant number. In terms of our parameters above, in this case a = 1, b = 2k, and c = k^2.

• In most exercises the quadratic equation has a = 1, and the equation becomes simply of the form x^2 + bx +c = 0. In these case, the denominator of the quadratic formula becomes 2a = 2*1 = 2. After a while, if you become careless you may forget that it is 2a, and not just 2 in the denominator, leading you to the wrong roots. To check your work, substitute each of the roots you found in turn instead of x in your quadratic equation's general formula and you should get to 0 = 0.

Resources

• Purplemath solving quadratic equations