We found that x=3 or x=- 23.
We can substitute our solutions back into the given equation and simplify to check if our answers are correct. We will start with x=3.
Again, we created a true statement. x=- 23 is indeed a solution of the equation.
b We want to solve the given equation for x. To do this, we can start by using factoring. Then, we will use the Zero Product Property.
Factoring
To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2-7x+10=0
In this case, we have 10. This is a positive number, so for the product of the constant terms in the factors to be positive, these constants must have the same sign (both positive or both negative.)
Factor Constants
Product of Constants
1 and 10
10
-1 and -10
10
2 and 5
10
-2 and -5
10
Next, let's consider the coefficient of the linear term.
x^2-7x+10=0
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, - 7.
Factors
Sum of Factors
1 and 10
11
-1 and -10
-11
2 and 5
7
- 2 and - 5
- 7
We found the factors whose product is 10 and whose sum is - 7.
x^2-7x+10=0
⇕
(x-2)(x-5)=0
Zero Product Property
Since the equation is already written in factored form, we can now use the Zero Product Property.
Again, we created a true statement. x=5 is indeed a solution of the equation.
c We want to solve the given equation for x. To do this, we can start by using factoring. Then, we will use the Zero Product Property.
Let's start factoring by identifying the greatest common factor (GCF).
Factor Out The GCF
The GCF of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the GCF is 2.
The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of 1.
2( x^2+x-6)=0
Let's temporarily only focus on this trinomial, and we will bring back the GCF after factoring.
Factor The Expression
To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2+x- 6=0
In this case, we have -6. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign (one positive and one negative.)
Factor Constants
Product of Constants
- 1 and 6
- 6
1 and -6
- 6
- 2 and 3
- 6
2 and -3
- 6
Next, let's consider the coefficient of the linear term.
x^2+1x- 6=0
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 1.
Factors
Sum of Factors
- 1 and 6
5
1 and -6
-5
- 2 and 3
1
2 and -3
- 1
We found the factors whose product is - 6 and whose sum is 1.
x^2+1x- 6=0
⇕
(x-2)(x+3)=0
Wait! Before we finish, remember that we factored out a GCF from the original equation. To fully complete the factored equation, let's reintroduce that GCF now.
2(x-2)(x+3)=0
Zero Product Property
Since the equation is already written in factored form, we can now use the Zero Product Property.
Again, we created a true statement. x=- 3 is indeed a solution of the equation.
d We want to solve the given equation by factoring.
Applying the Difference of Two Squares
Do you notice that the expression
on the left-hand side of the equation is a difference of two perfect squares? This can be factored using the difference of squares method.
a^2 - b^2 ⇔ (a+b)(a-b)To do so, we first need to express each term as a perfect square.
Expression
4x^2-1
Rewrite as Perfect Squares
(2x)^2 - 1^2
Apply the Formula
(2x+1)(2x-1)
Solving the Equation
Finally, to solve the equation, we will use the Zero Product Property.
