In elementary algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form
where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, the quadratic coefficient, the linear coefficient and the constant or free term.
Solving the quadratic equation
A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.
Factoring by inspection
It may be possible to express a quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.
Completing the square
The process of completing the square makes use of the algebraic identity
which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, ax2 + bx + c = 0 1. Divide each side by a, the coefficient of the squared term. 2. Rearrange the equation so that the constant term c/a is on the right side. 3. Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square. 4. Write the left side as a square and simplify the right side if necessary. 5. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side. 6. Solve the two linear equations.
We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0
Quadratic formula and its derivation
Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The mathematical proof will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:
Taking the square root of both sides, and isolating x, gives:
Reduced quadratic equation
It is sometimes convenient to reduce a quadratic equation to an equation involving two instead of three constant coefficients. This is done by simply dividing both sides by a, which is possible because a is non-zero. This produces the reduced quadratic equation
Here p = b/a and q = c/a are the only coefficients in the reduced equation, which is also called a monic equation. It follows from the quadratic formula that the solution to the reduced quadratic equation is
In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta
A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases: If the discriminant is positive, then there are two distinct roots
If the discriminant is zero, then there is exactly one real root
If the discriminant...
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