Equations and Solutions to Equations An equation is a statement that two algebraic expressions are equal. To solve an equation in x means to find all values of x for which the equation is true. Isolate x.(i.e. Get x by itself.) When solving an equation involving fractions, clear the fractions by multiplying by the Least Common Denominator (LCD)
Intercepts of a Graph The x -intercept is the point at which the graph crosses the x -axis. (a, 0) Let y = 0, and solve for x. The y -intercept is the point at which the graph crosses the y -axis. (0, b) Let x = 0, and solve for y.
Intercepts and Solutions In order to solve an equation, set y = 0, which is the same process for finding the x -intercept. So, the x -intercepts of the graph ARE the solutions to the equation.
Points of Intersection of Two Graphs An ordered pair that is a solution of two different equations is a point of intersection. To solve for the point of intersection of two equations, solve one equation for one variable and substitute that expression into the other equation. Set them equal to one another if solved for y.
Extraneous Solutions??? When multiplying or dividing by a variable expression, it is possible to introduce an extraneous solution. This is a solution that DOES NOT satisfy the original equation. Ignore If a solution makes a denominator zero, is it extraneous.
Polynomial Equations Polynomial equations are classified by their degree (the greatest power of the variable): First degree - linear equation Second degree - quadratic equation Third degree - cubic equation
Solving Quadratics The quadratic formula can ALWAYS be used to solve a quadratic equation. First, get the equation in the form:
Solving Polynomial Equations by Factoring Get all terms on one side of the equation set equal to zero. Factor out the G.C.F. FIRST Now factor the resulting polynomial using the various rules. Difference of perfect squares Sum or difference of perfect cubes “Reverse FOILing” of trinomials