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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You can solve any quadratic equation by completing the square.

Now we will complete the square to solve ax2 + bx + c = 0. The solutions will be expressed in terms of a, b, and c. These solutions will give us a formula we can use to solve any quadratic equation.

Step 1 Isolate the x2-term and the x-term on one side of the equation.

Subtract c from both sides of the equation.

 ax2 + bx + c = 0

ax2 + bx = -c

Step 2 If the coefficient of x2 is not 1, divide both sides of the equation by the coefficient of x2.

The coefficient of x2 is a.

Divide both sides of the equation by a.
Step 3 Find the number that completes the square: Multiply the coefficient of x by . Square the result.

The coefficient of the x-term is .

Step 4 Add the result of Step 3 to both sides of the equation.

Add to both sides of the equation.

To combine like terms on the right side, write both fractions with denominator 4a2.

Combine like terms on the right side. In the numerator, write the b2-term first.

Step 5 Write the trinomial as the square of a binomial.

Step 6 Finish solving using the Square Root Property.

Use the Square Root Property. Rather than writing two separate equations, we write a single equation using the ± sign.
Subtract from both sides and simplify the radical.
Combine the fractions into a single fraction.


If a > 0, then 4a2 = 2a.

If a < 0, then 4a2 = -2a.


The result is called the quadratic formula.


Formula — The Quadratic Formula

The solutions of the quadratic equation ax2 + bx + c = 0 are given by the quadratic formula:

Here, a, b, and c are real numbers and a 0.