A Complete Guide to Solving Quadratic Equations

A rug with the quadratic formula.
A rug with the quadratic formula.

Quadratic Equations

First, what is a quadratic equation?

3x^2+4x=0 Yes, the highest power is 2 (for 3x^2).

-4x^2=0 Yes, the highest power is 2 (for -4x^2).

9x+3=0 No, the highest power is not 2.

The format of a quadratic equation is:

ax^2 + bx + c = 0, where a, b and c are real numbers and a is not 0. If a is zero, the equation is really just bx + c=0, and there is not second power. In fact quadratic really refers to the degree of the equation. Degree means the highest power you see in the equation, which must always be 2 for a quadratic equation.

Quadratic equations have two solutions, which may or may not be real. This article will address three different ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula (in order of difficulty). Only the quadratic formula will help you find solutions that are not real (imaginary).

Solve by factoring:

Example: x^2 = 16

First write the equation in standard quadratic form ax^2+bx+c=0 by subtracting 16 from both sides: x^2 - 16 = 0. Here, a=1, b=0, c=-16. Then comes the tricky step and might take a bit of practice at first. We try a pair of number that together multiply to the "c" term (which is -16 in this case). This pair also needs to add up to the "b" term (0). In this case 4 and -4 work: (x + 4)(x - 4) = 0. We then set each factor equal to 0: (x + 4) = 0 or (x - 4) = 0. We can then solve to find that: x = -4 or x = 4.

Solve by completing the square:

Example: x^2 = 16. First, write the equation in standard quadratic equation form by subtracting 16 from both sides: x^2 - 16 = 0. Apply the square root to both sides: x = +/- sqrt(16). Thus the answer is : x = +/- 4.

Harder Example: x^2+4x+3=0. Here a=1, b=4, c=3. We must rewrite the equation as (x+b)^2=-c+b^2. In our case, we have (x+2)^2=-3+4. This simplifies to (x+2)^2=1. We can take the square root of both sides to find that (x+2)= 1 or -1. We solve each part individually to find that x=-1 or -3.

Solve by using the quadratic formula:

Example:3x^2 + 16x + 5 = 0. This example is already written in the standard quadratic equation form; therefore, we know that a = 3, b = 16 and c = 5. Substitute the values for a, b and c into the quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). In our case, x = (-16 +/- sqrt(16^2 - 4*3*5)) / (2*3). Thus x = (16 - 14) / 6 or x = (16 + 14) / 6. Further simplifying results in finding x = -1/3 or x = -5.

Happy solving!


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