## Unit 3: Standard Form

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.

## Learning Goals

• Be able to complete the square to get the vertex form equation.
• 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.

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.

## 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.

## 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:

## 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: