A quadratic equation is an equation that has a second-degree term and no higher terms. A second-degree term is a variable raised to the second power, like x2. When you graph a quadratic equation, you get a parabola, and the solutions to the quadratic equation represent where the parabola crosses the x-axis. A quadratic equation can be written in the form:

quadratic equation,

where a, b, and c are numbers (a ≠0), and x is the variable. x is a solution (or a root) if it satisfies the equation ax2 + bx + c = 0.

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Some examples of quadratic equations include:

3×2 + 9x – 2 = 0 6×2 + 11x = 7 4×2 = 13

——————————————————————————– Solving a Quadratic Formula:

Some quadratic equations can be solved easily by factoring. Some simple-to-solve quadratic equations are:

x2 – 1 = 0

(x + 1)(x – 1) = 0

x = ±1

x2 – 5x + 6 = 0

(x – 2)(x – 3) = 0

x = 2, 3

Most second-degree equations are more difficult to solve, and cannot be solved by simple factoring. The quadratic formula is a general way of solving any quadratic equation:

quadratic formula

This formula gives two solutions, although the two solutions may be the same

number. (When solving any polynomial equation of degree n, there are at most n solutions to that equation.)

——————————————————————————– Deriving the Quadratic Formula:

The quadratic formula is obtained by solving the general quadratic equation. This is one way to derive the quadratic formula: Deriving Quadratic

Divide each side of the equation by a.

Deriving Quadratic

Subtract c/a from each side of the equation.

Deriving Quadratic

Add (b/2a)2 to each side of the equation (to complete the square). Deriving Quadratic

Find a common denominator for the right side of the equation. Deriving Quadratic

Take the square root of each side of the equation.

Deriving Quadratic

Add b/2a to each side of the equation.

Deriving Quadratic

The plus-or-minus sign shows that there are two possible solutions.

——————————————————————————–

parabolas

•The green parabola has 2 x-intercepts. Its corresponding quadratic equation has 2 distinct solutions (x=1 and x=4). •The yellow parabola has 1 x-intercept. Its corresponding quadratic equation has 1 solution (x=-3). •The purple parabola has no x-intercepts. Its corresponding quadratic equation has no solutions.

You can try to solve any quadratic equation by using the quadratic formula (or by solving it algebraically, similar to the way we just derived the

quadratic formula above). Every quadratic equation has at most two solutions, but for some equations, the two solutions are the same number, and for others, there is no solution on the number line (because it would involve the square root of a negative number).

One way to understand this visually is to realize that when you graph a quadratic equation (a second-degree equation), you get a parabola. When you set the quadratic equation equal to zero, this represents the points where the parabola hits the x-axis (the x-intercepts, where y=0).

——————————————————————————– Introduction to Algebra – Printable Worksheets

Quadratic

Quadratic Equations Worksheets (with no linear term)

Solve simple (pure) quadratic equations (no linear terms).

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