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| 11 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
A quadratic equation is a polynomial equation of degree 2. There is a special name for quadratic equations whose linear coefficient b is 0. These equations can be written in the form ax^2+c=0 and have their own characteristics.
If the linear coefficient b of a quadratic equation is 0, the equation is called a simple quadratic equation and can be written in the following form.
ax^2+c=0
This type of equation can be solved using inverse operations. Once x^2 is isolated, the equation can be written as x^2=d, where d=- ca. The value of d gives the number of solutions the equation has.
d>0:& 2real solutions d=0:& 1real solution d<0:& 0real solutions
The cases d>0, d=0, and d<0 will be discussed one at a time.
sqrt(a^2)=± a
State solutions
If d=0, then the equation x^2=d can be written as x^2=0. This equation can be solved for x. x^2=0 ⇔ x* x=0 By using the Zero Product Property, it can be concluded that x=0. This is the only solution for the equation.
Because the square of any real number is always greater than or equal to 0, if d<0 the equation x^2=d has no real solutions.
Heichi is going on a trip with a friend. He wants to finish up his homework first, so he does not have to worry about it when he gets home.
Write the equations in the form x^2=d. If d>0, the equation has two real solutions. If d=0, then the equation has one real solution. Finally, if d<0, the equation has no real solutions.
Equation | Rewrite as x^2=d | Value of d | Number of Real Solutions |
---|---|---|---|
-4x^2+5=5 | x^2= 0 | d= 0 | One |
5x^2-125=0 | x^2= 25 | d= 25 ⇒ d>0 | Two |
- 3x^2-27=0 | x^2= - 9 | d= - 9 ⇒ d<0 | Zero |
Without solving the simple quadratic equations, determine the number of real solutions.
Apart from determining the number of real solutions of a simple quadratic equation, most of the times it is important to calculate those solutions.
Simple quadratic equations are quadratic equations whose linear coefficient b is equal to 0. ax^2+c=0 This type of equation can be solved using inverse operations, and two steps must be followed.
sqrt(LHS)=sqrt(RHS)
sqrt(a^2)=± a
Calculate root
State solutions
Ali and Heichi are enjoying a ski vacation.
Start by isolating x^2.
sqrt(LHS)=sqrt(RHS)
sqrt(a^2)=± a
sqrt(a/b)=sqrt(a)/sqrt(b)
Calculate root
State solutions
Start by isolating x^2.
sqrt(LHS)=sqrt(RHS)
sqrt(a^2)=± a
State solutions
(I), (II): Use a calculator
(I), (II): Round to 3 significant digit(s)
Solve the following simple quadratic equations by taking square roots. If necessary, round the solutions to two decimal places.
Jordan is representing North High School in an algebra competition.
Start by isolating (x-5)^2.
The quadratic equation given in the last example had a specific format. Equation:& - 2(x-5)^2+2=0 Format:& a(x-h)^2+k=0 It is worth noting that all quadratic equations can be written in this format by a process called completing the square.
Equation | Rewrite |
---|---|
- x^2+4=0 | - 1(x-0)^2+4=0 |
3x^2-6x+5=0 | 3(x-1)^2+2=0 |
5x^2=3x+1 | 5(x-3/10)^2+(- 29/20)=0 |
4x^2+4x=- 2 | 4(x-(- 1/2))^2+1=0 |
4x^2+4x=- 2 | 4(x-(- 1/2))^2+1=0 |
10x^2+160=80x | 10(x-4)^2+0=0 |
Find the real roots, or solutions, of the following quadratic equations.
The equation can be solved by taking square roots on both sides. Remember that to ensure we find all potential solutions, we need to consider both the principal root and the negative root!
Therefore, the equation has two real solutions. These solutions are x=8 and x=-8.
This time we need to isolate the x^2-term before taking square roots on both sides. Once again, if we do not want to miss any solution, we need to consider both the principal root and the negative root.
Therefore, the equation has two real solutions. These solutions are x=3 and x=-3.
Consider the following rectangle.
We begin by recalling that the area of a rectangle can be found by multiplying its length l by its width w. A=l w In the diagram we can see the dimensions of the rectangle. Let's substitute them into the formula!
The area of the rectangle is 6x^2 square units.
We are told that the area of the rectangle is 54 square units. Let's substitute this value into the expression we wrote in Part A and solve for x.
Since a negative length does not make sense, we can discard the negative solution and only consider the principal root. Therefore, the answer is x=3.
Find the real roots, or solutions, of the following quadratic equations.
We will begin by isolating the x^2-term of the equation. Then we will take square roots on both sides. Remember that to find all of the potential solutions to the equation, we will consider both the principal root and the negative root.
Therefore, the equation has two real roots. These roots are x=2 and x=-2.
Let's see what happens if we isolate the x^2-term of the equation and take square roots on both sides.
The square root of a negative number has no real solution. This means that the equation has no real roots.
Find the real roots, or solutions, of the following quadratic equations.
The given equation involves the square of a binomial. However, we do not need to expand it. Instead, we will take the square root on both sides of the equation and continue to simplify from there.
Therefore, the roots, or solutions, of the equation are x=4 and x=-2.
Let's solve this equation the same way we did in Part A. We will start by taking the square roots of both sides of the equation, then simplify as much as possible.
Therefore, the roots of the equation are x=2 and x=-6.