Quadratic Relationships

By: Guneet Jaswal

Unit 3: Standard Form

What is Quadratics?

Quadratics is the part of algebra which requires you to work with quadratic equations. A quadratic equation is an equation in which the highest exponent of the variable is a 2 (a square). When a quadratic relation is graphed it forms a parabola which is a type of curve in which each point is located at an equal distance from the point that is opposite to it.

Standard Form Equation: y=ax^2+bx+c

Learning Goals

  • Be able to complete the square to get the vertex form equation.
  • Solve quadratic equations using the quadratic formula.
  • Be able to find the number of x-intercepts by looking at the value of the discriminant.
  • Solve word problems

Summary of Standard Form

The first thing you need to know in this unit is what the variables a and c are in the equation. The a value in this equation gives you the shape and the direction of opening of the parabola. The c value is the y-intercept of the parabola. You also need to know how to complete the square to get an equation into vertex form. This can allow you to determine a maximum or minimum value for any parabola. The procedure of completing the square requires you to change the first two terms of a quadratic equation in standard form into a perfect square while maintaining the balance of the original relation. Another thing you need to know is how to use use the quadratic formula to solve quadratic equations. In order to do this all you need to do is substitute the values into the equation. Along with this you need to be able to solve word problems using the quadratic formula or completing the square. This unit has many things you need to learn and they can be very easy once you understand them.

Quadratic Formula

The quadratic formula is x = [ -b ± sqrt(b^2 - 4ac) ] / 2a.


How to solve equations using the quadratic formula:

  • Determine the a,b and c values in the equation
  • Plug these values into the equation
  • Solve the equation to find x.


Example: Use the quadratic formula to solve the equations below.

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The Discriminant

The number inside the square root of the quadratic formula is called the Discriminant (D). It allows us to determine the number of solutions the quadratic equation has.


D=b^2-4ac


  • If the discriminant is greater than 0 than there are 2 solutions.
  • If the discriminant is smaller than 0 than there are no solutions.
  • If the discriminant is 0 than there is 1 solution.

Completing the Square

Completing the square is the process in which an equation is converted from standard form to vertex form.


y=ax^2+bx+c -> y=a(x-h)^2+k


Completing the square can be used to find a maximum or minimum point or to solve a maximum revenue word problem.


How to complete the square:


  • Put brackets around the first two terms in the equation. If the a value is greater than one take it out of the brackets and divide it out of the rest of the terms in the brackets.
  • Next, divide the b value in the equation by 2.
  • Then, square that value.
  • Plug the value you get in the previous step into the equation. This value goes in the brackets after the second term. The number must first be added and then subtracted.
  • Then, take the negative value that was added in the brackets outside of them. If there is an a value outside of the brackets, make sure you multiply this value with the a value.
  • Find the perfect square of the equation in brackets and simplify the values outside of the brackets.
  • Now you have your equation in vertex form and you have completed the square.


Example: Rewrite the equation in the form y=a(x-h)^2+k by completing the square.

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Word Problem Using the Quadratic Formula

Here is an example of a word problem using the quadratic formula:


The length of a rectangle is 16cm greater than its width. The area is 35m^2. Find the dimensions of the rectangle to the nearest hundreth of a metre.


Solution to the word problem:

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Word Problem Using Completing the Square

Here is an example of a word problem using completing the square:


The path of a ball is modeled by the equation y=-x^2+2x+3, where x is the horizontal distance, in metres, from a fence and y is the height, in metres above the ground.


a) What is the maximum height of the ball, and at what horizontal distance does it occur?

b) Sketch a graph to represent the path of the ball.


Solutions to the problem:

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Reflection

I learned a lot of new things while working on the quadratics unit with vertex form, factored form and standard form equations. I learned how to graph the quadratic relations, determine transformations, create equations, factor equations, solve word problems and so much more. I feel that I am pretty good at the quadratics unit as I did really well on my tests and tips. My favourite unit of quadratics was standard form since it was fairly simple and not a lot to do. It mainly required us to complete the square or use the quadratic formula. There were also many word problems but they were easy to understand. Second would be the factored form unit because we had to factor equations using the different methods and I found that easy to do. We also had to graph the relations, solve word problems and expand and simplify polynomials. The word problems were pretty similar so it was easy to do and there was not much graphing. Third would be vertex form because it was the first unit and it was kind of confusing in the beginning. It also required us to graph a lot and I don't like to graph that much so I did not like that part. It was still pretty easy to determine transformations and write the equation in vertex form. Also after this unit I started recognizing parabolas more often in my everyday life. While working on these units I found that they all related with each other somehow. In the standard form unit we had to complete the square to change the standard form equation to vertex form and when doing factored form we had to graph the parabolas just like we did with vertex form. Another thing is that all the word problems in each unit were very similar to each other. For example the flight path of an object question. All units required us to find x-intercepts and the vertex of a parabola as well. Another thing is that the equations can all be converted to be in each form and can be used to solve different problems in different units. The equations all had the variable a in common as well which helped determine the shape and direction of opening of the parabola. Overall, I found this quadratics unit easy and I enjoyed it. Although things may have been confusing at times and I made some mistakes I continued to practice the questions and started doing a lot better. Also, I did very well on my tests so I believe that this is a topic I am good at. In conclusion, I learned a lot about quadratics and it was a unit I liked.
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