Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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When you solve a quadratic equation when you have been given a y-value and need to find all of the corresponding x-values. For example, if you had been given the quadratic equation:

y = x2 + 8 · x +10,

and the y-value,

y = 30,

then solving the quadratic equation would mean finding all of the numerical values of x that work when you plug them into the equation:

x2 + 8 · x +10 = 30.

Note that solving this quadratic equation is the same as solving the quadratic equation:

x2 + 8 · x +10 - 30 = 30 - 30 (Subtract 30 from each side)
x2 + 8 · x - 20 = 0 (Simplify)

Solving the quadratic equation x2 + 8 · x - 20 = 0 will give exactly the same values for x that solving the original quadratic equation, x2 + 8 · x +10 = 30, will give.

The advantage of manipulating the quadratic equation to reduce one side of the equation to zero before attempting to find any values of x is that this manipulation creates a new quadratic equation that can be solved using some fairly standard techniques and formulas.

Solving a polynomial equation is exactly the same kind of process as solving a quadratic equation, except that the quadratic might be replaced by a different kind of polynomial (such as a cubic or a quartic).


The Number of Solutions of a Polynomial Equation

A quadratic is a degree 2 polynomial. This means that the highest power of x that shows up in a quadratic’s formula is x2. The maximum number of solutions that a quadratic function can possibly have is 2.

The maximum number of solutions that a polynomial equation can have is equal to the degree of the polynomial.

It is possible for a polynomial equation to have fewer solutions (or none at all). The degree of the polynomial gives you the maximum number of solutions that are theoretically possible, not the actual number of solutions that will occur.

Example: Solving a Polynomial Equation Graphically

The graph given below shows the graph of the polynomial function:

Use the graph to find all solutions of the polynomial equation:


Graphically, as the polynomial equation is equal to zero the solutions of the polynomial equation,

will be the x-coordinates of the points where the graph of the polynomial:

touches or crosses the x-axis. If you look carefully at the graph supplied above, the graph of the polynomial touches or cuts the graph at the following points:

x = -2, x = -1, x = 2.

The solutions of the polynomial equation

are x = -2, x = -1 and x = 2.