The standard form of a quadratic equation is:
ax2 + bx + c = 0, where a ≠ 0
In the equation, a, b, and c are constants, and x is a variable. The degree of the equation, 2 (the exponent on x), makes the equation quadratic. Quadratic equations of this form can be solved for x to find the roots of the equation, which are the point(s) where the equation is equal to 0. Roots can also be referred to as zeros.
Solving quadratic equations
There are a number of different methods for solving a quadratic equation. Below are several of them.
Quadratic equations of the form ax2 + c = 0
A quadratic equation without the x1 term is relatively simple to solve. We don't need to factor or use the quadratic formula (discussed later). All we need to do is isolate x as we would when trying to solve for x in any equation, then take the square root of a constant.
Given x2 - 4 = 0, solve for x:
x2 = 4
x = ± = ± 2
One of the key things we need to remember when solving quadratic equations is that x can take on both positive and negative values, since both -2 × -2 and 2 × 2 = 4. this also means that if bot a and c is positive or negative, there are no real solutions since it is not possible to take the square root of a negative number without using imaginary numbers.
Solving equations using factoring relies on using one of the properties of 0. If the product of two numbers, or expressions, is 0, then at least one of the expressions must equal 0. This allows us to separate the factors and set them equal to 0 individually to find the solution(s) to the equation.
1. Solve 2x2 - 8x = 0:
2x(x - 4) = 0
We can separate this and solve for 2x = 0 and x - 4 = 0:
2x = 0
x = 0
x - 4 = 0
x = 4
The equation has two solutions, x = 0 and x = 4.
2. Solve x2 - 4x + 4 = 0:
x2 - 4x + 4 = (x - 2)2 = 0
x - 2 = ± 0
x = 2
In this case, even though we are taking a square root, 0 is neither positive nor negative, so there is only one solution. This will always be the case in equations that can be factored into the form (x - c)2, so once you start recognizing these equations in their expanded form, x2 - 2cx + c2, you'll be able to solve them relatively quickly.
3. Solve x2 - x - 6 = 0:
x2 - x - 6 = (x - 3)(x + 2) = 0
x - 3 = 0
x = 3
x + 2 = 0
x = -2
The two solutions to the equation are x = 3 and x = -2.
Using the quadratic formula
The terms "quadratic formula" and "quadratic equation" are sometimes used interchangeably, but the two should not be confused. The quadratic formula refers specifically to a formula used to solve quadratic equations:
The quadratic formula can be thought of as a "brute force" method for solving quadratic equations since it can be used to solve any quadratic equation in standard form, like all of the examples above. Depending on the specific quadratic equation however, it is often simpler to use a method such as factoring, completing the square, or some other method where possible, before using the quadratic formula. That being said, the quadratic formula itself is relatively simple to use as long as the equation is in standard form.
The a, b, and c in the quadratic formula are all constants, and refer to the coefficients of the standard form of a quadratic equation:
ax2 + bx + c
Solving a quadratic equation using the quadratic formula just requires us to plug the coefficients of the equation into the formula.
Solve 7x2 - 13x + 6 = 0:
In the above equation, a = 7, b = -13, and c = 6. Plugging these into the quadratic formula:
x = and x =
Though the quadratic formula is tedious, it is very powerful in that it enables us to solve any quadratic equation as long as we put it in standard form.
In the quadratic formula, the expression underneath the square root sign, b2 - 4ac, is known as the discriminant. It is worth noting that if:
b2 - 4ac = 0, there is only one solution
b2 - 4ac > 0, there are two real solutions
b2 - 4ac < 0, there are no real solutions
One other method for solving quadratic equations is completing the square.