Solving Quadratic Equations

By Factoring

Firstly, when solving a quadratic equation by factoring, the equation needs to be in standard form(y=ax²+bx+c). There will often be no common factor of the equation, which means you will have to use decomposition. To do this, you will find two numbers that give you the product of 'a' and 'c', and the sum of 'b'. For example, for the equation x²+3x+2=0, the sum is 3 and the product is 2 (as 1x2=2). Two numbers that multiply to 2 and add to 3 are 1 and 2. Next you decompose 'b', so in his case 3, and turn it into 2x+x. This will make your equation look like this: x²+2x+x+2=0. Now you can factor by grouping, and will end up with (x+1)(x+2)=0. Finally, you solve for x in the brackets, and will get x=-1 and x=-2.

By Completeing the Square

The best way to learn this method is by using an example.



1. Block off the first two terms→ (-5x²+20x)-2=0

2. Factor out the A→ -5(x²-4x)-2=0

3. Add "zero" by: dividing the middle term by 2, and squaring it→ -5(x²-4x+4-4)-2=0

4. Bring out the negative→-5(x²-4x+4)+20-2=0

5. Factor inside the brackets and simplify→-5(x-2)²+18=0

6. Divide equation by the number in front of the brackets to get rid of it→(x-2)²-18/5=0

7. Bring the second term over to the other side, and square root all to get rid of the square on the first term→√(x-2)²=√18/5

8. Isolate x, and simplify→x=±3.6+2

9. You will get two answers because the number that was square rooted is positive and negativex=5.6 or x=-1.6

By Using the Quadratic Formula

To solve an equation using the quadratic formula, the equation needs to be in standard form, and one side should equal zero. Then you take your a, b and c values, and plug them into the quadratic formula. The video below will show you the quadratic formula and how to use it.
Solving Quadratic Equations using the Quadratic Formula - Example 3

Properties of Quadratic Relations

Important Terms and Characteristics

Parabola→the graph of a quadratic relation

Vertex→the point on the graph with the greatest y-coordinate (if the graph opens down) or the least y-coordinate (if the graph opens up); vertex is directly above or below the midpoint of the x-intercepts (if the parabola crosses the x-axis)

Optimum Value→the y-coordinate of the vertex corresponds to the optimum value, either a maximum or minimum value

Minimum Value→ y-coordinate of the vertex when the graph opens up

Maximum Value→ y-coordinate of the vertex when the graph opens down

Direction of Opening→ if the constant value of the second differences is positive, the parabola opens up; and if its is negative the parabola opens down

Axis of Symmetry→ a vertical line that passes through the vertex (perpendicular bisector) that makes the parabola symmetrical

Equation of the Axis of Symmetry→ As the coordinates of the vertex are (h,k), the equation would be x=h since the line passes through the vertex

Zeros/x-intercepts→ the x-coordinates of the points the parabola crosses on the x-axis

Graphing Factorable Quadratic Equations

Three Things to Consider When Graphing

1. x-intercept(s): Factor the equation and determine the x-intercepts whenever possible

2. y-intercept: set x=0, and determine this point

3. vertex: the maximum or minimum point of the parabola; very important when solving optimization questions

The video below will demonstrate how to find these three things when your equation is factored, and use them to graph your quadratic equation.

Stretching, Reflecting and Translations of Quadratics

Parent/Original/Base Parabola

The most simple parabola is y=x². The graph passes through the origin (0,0), and is contained in Quadrants I and II. All other parabolas, or quadratic functions, can be obtained from this graph by one or more transformations.


Vertical Stretch

- parabola gets thinner

- y-values increase

- if the A value is greater than 1

Vertical Compression

- parabola becomes wider

- y-values decrease

- if the A value is less than 0 and greater than 1 or less than -1 and less than 0 (fractions)


When the A value is a negative, it means that the parabola opens down. This mean the parabola would be reflecting across the a-axis.


In y=ax², the A produces a transformation. Additionally, in y=a(x-h)²+k , the h and k values produce translations.

In y=x²+k:

- the "k" produces a vertical translation up or down

- the parabola does not stretch or compress (as there is no "a" value)

- affects the y-values

In y=(x-h)²:

- the "h" horizontally translates the parabola left or right

- affects the x-values

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Different Forms of Quadratic Equations

Standard Form


A value:

-if the parabola opens up or down

-vertical stretch or compression

C value:


Factored Form


A value:

-opens up or down

-vertical stretch or compression

R and A values:


Vertex Form


A value:

-opens up or down

-vertical stretch or compression

H value:

-x-coordinate of vertex

-horizontal shift

-axis of symmetry

K value:

- y-coordinate of vertex

- vertical shift

- optimum value

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Word Problems

When solving quadratic word problems, there are two possible types you could be working with; optimizing or variable type.

Optimizing Type

When optimizing, you are looking for the maximum, minimum, biggest, smallest, etc. You are not solving for 0. you would be solving for a variable. Within optimizing, you can either have a one equation question, or a two equation question. If it is one equation, you would simply find the vertex by completing the square or finding the middle of your roots. If it is two equations, you would solve for the variable by isolating it, and using substitution.

Variable Type

In variable type, one side will equal to zero, and you will be given one veriable and asked to solve for the other. Similarly to optimizing type, you can either have a one equation question, or a two equation question. When you are given only one equation, you would make one variable in the equation have the value of zero, to be able to solve for the other variable. In a two equation question, you can also make one variable in the equation have the value of zero to solve for the other variable, or you can use substitution.