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Quadratic equation

Revision as of 21:55, 17 June 2006 by IntrepidMath (talk | contribs) (Copied From completing the Square for completeness.)

Quadratic Equations

A quadratic equation is an equation of form ${a}{x}^2+{b}{x}+{c}=0$. a, b, and c are constants, and x is the unknown variable. Quadratic Equations are solved using 3 main strategies, factoring, completing the square, and the quadratic formula.

Factoring

The purpose of factoring is to turn a general quadratic into a product of binomials. This is easier to illustrate than to describle

Example: Solve the equation $x^2-3x+2=0$ for x. Solution: $x^2-3x+2=0$ First we expand the middle term. This is different for all quadratics. We cleverly choose this so that it has common factors. We now have $x^2-x-2x+2=0$ Next, we factor out our common terms to get: $x(x-1)-2(x-1)=0$. We can now factor the (x-1) term to get: $(x-1)(x-2)=0$. By a well know theorem, Either $(x-1)$ or $(x-2)$ equals zero. We now have the pair or equations x-1=0, or x-2=0. These give us answers of x=1 or x-2. Plugging these back into the original equation, we find that both of these work! We are done.

Completing the square

Completing the square

Quadratic Formula

General Solution For A Quadratic by Completing the Square

Let the quadratic be in the form $a\cdot x^2+b\cdot x+c=0$.

Moving c to the other side, we obtain

$ax^2+bx=-c$

Dividing by ${a}$ and adding $\frac{b^2}{4a^2}$ to both sides yields

$x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}$.

Factoring the LHS gives

$\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$

As described above, an equation in this form can be solved, yielding

${x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}$

This formula is also called the Quadratic Formula.

We simply plug in a, b, and c and out pops the 2 values of x.

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