To solve equations through factorization, we must transpose all the elements to one side of the equation and leave on the other side.
Why? Because after factoring, we will have as the product.
To solve equations through factorization, we must transpose all the elements to one side of the equation and leave on the other side.
Why? Because after factoring, we will have as the product.
The product of two numbers equals when, at least, one of them is .
If
then
either:
or:
or both are equal to .
Solve for x.
\( -x^2-7x-12=0 \)
Find the value of the parameter x.
\( (x-5)^2=0 \)
Find the value of the parameter x.
\( x^2-25=0 \)
Find the value of the parameter x.
\( 2x^2-7x+5=0 \)
Find the value of the parameter x.
\( (x-4)^2+x(x-12)=16 \)
Solve for x.
First, factor using trinomials and remember that there might be more than one solution for the value of :
Divide by -1:
Therefore:
Or:
Find the value of the parameter x.
We will factor using the shortened multiplication formulas:
Let's remember that there might be more than one solution for the value of x.
According to one solution, we'll take the square root:
According to the second solution, we'll use the shortened multiplication formula:
We'll use the trinomial:
or
Therefore, according to all calculations,
Find the value of the parameter x.
We will factor using the shortened multiplication formulas:
Remember that there might be more than one solution for the value of x.
According to the first formula:
We'll take the square root:
We'll take the square root:
We'll use the first shortened multiplication formula:
Therefore:
Or:
Find the value of the parameter x.
We will factor using trinomials, remembering that there is more than one solution for the value of X:
We will factor -7X into two numbers whose product is 10:
We will factor out a common factor:
Therefore:
Or:
Find the value of the parameter x.
Let's open the parentheses, remembering that there might be more than one solution for the value of X:
Therefore:
Or:
Find the value of the parameter x.
\( -2x(3-x)+(x-3)^2=9 \)
Find the value of the parameter x.
\( (x+5)^2=0 \)
Find the value of the parameter x.
\( 12x^3-9x^2-3x=0 \)
A right triangle is shown below.
\( x>1 \)
Calculate the lengths of the sides of the triangle.
A right triangle is shown below.
\( x>1 \)
Find the lengths of the sides of the triangle.
Find the value of the parameter x.
To solve the equation , follow these steps:
Therefore, the values of that satisfy the equation are and .
Find the value of the parameter x.
To solve the equation , we will use the fact that a perfect square is zero only when the quantity being squared is zero itself.
Therefore, the solution to the equation is .
Find the value of the parameter x.
To solve the problem, we follow these steps:
Let's work through the solution:
Step 1: Observe that each term in the equation has a common factor of . So, we can factor out of the equation, giving us:
Step 2: Having factored out , we now have a product of terms equaling zero. According to the zero-product property, at least one of the factors must be zero:
This gives us one solution directly:
Step 3: Solve the quadratic equation using the quadratic formula, where , , and :
The quadratic formula is:
Applying it to our equation:
This gives us two solutions:
When , .
When , .
Therefore, the solutions to the equation are , , and .
Verifying against the provided choices, the correct choice is choice 2: .
A right triangle is shown below.
x>1
Calculate the lengths of the sides of the triangle.
To find the lengths of the sides of the right triangle, we will apply the Pythagorean theorem, which states for a right triangle, where is the hypotenuse.
Given the side lengths are , , and , we assume is the hypotenuse because it is the largest value and confirm it by checking with the theorem.
Substitute into the Pythagorean theorem:
Let's expand each side and solve for :
Combine these into a single equation:
Simplify and combine like terms:
Rearrange to form a quadratic equation:
Factor the quadratic equation:
Solve for :
(Not valid as )
Therefore, substituting will provide the side lengths:
- Short side:
- Other side:
- Hypotenuse:
These side lengths , , and form a well-known Pythagorean triple. Therefore, the solution to the problem is .
A right triangle is shown below.
x>1
Find the lengths of the sides of the triangle.
To solve this problem, we begin by using the Pythagorean theorem, as the triangle is right-angled. Let's identify the hypotenuse:
Using the Pythagorean theorem, we write:
Let's expand and simplify the equation:
Simplifying further:
Rearrange all terms to one side:
Simplifying gives:
This is a standard quadratic equation, which we can solve using factoring. By factoring, we find:
Setting each factor equal to zero gives solutions and . Since , we discard .
The valid solution is .
Now, substitute back into the expressions for the side lengths:
Therefore, the lengths of the sides of the triangle are , , and , which matches choice 4.
Therefore, the correct answer is choice 4: .
In front of you is a square.
The expressions listed next to the sides describe their length.
( \( x>-4 \) length measurements in cm).
Since the area of the square is 36.
Find the lengths of the sides of the square.
In front of you is a square.
The expressions listed next to the sides describe their length.
( \( x>-2 \) length measurements in cm).
Since the area of the square is 16.
Find the lengths of the sides of the square.
In front of you is an isosceles right triangle.
The expressions listed next to the sides describe their length.
( \( x>-5 \) length measurements in cm).
Since the area of the triangle is 12.5.
Find the lengths of the sides of the triangle.
In front of you is an isosceles right triangle.
The expressions listed next to the sides describe their length.
( \( x>-8 \) length measurements in cm).
Since the area of the triangle is 32.
Find the lengths of the sides of the triangle.
In front of you is a square.
The expressions listed next to the sides describe their length.
( x>-4 length measurements in cm).
Since the area of the square is 36.
Find the lengths of the sides of the square.
To solve for the side length of the square, we follow these steps:
.
Taking the square root of both sides,
or .
Only satisfies the condition .
Side length = cm.
Therefore, the length of the sides of the square is cm.
6
In front of you is a square.
The expressions listed next to the sides describe their length.
( x>-2 length measurements in cm).
Since the area of the square is 16.
Find the lengths of the sides of the square.
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: Given that the area of the square is 16, we use the area formula: .
Step 2: Solving the equation, take the square root of both sides:
This produces two solutions:
Step 3: Solve each equation for :
For , we have:
For , we have:
Since , we discard the solution because it does not satisfy the condition.
Thus, the acceptable value is .
The length of the sides of the square is cm.
Therefore, the solution to the problem is 4, and it matches with choice 1.
4
In front of you is an isosceles right triangle.
The expressions listed next to the sides describe their length.
( x>-5 length measurements in cm).
Since the area of the triangle is 12.5.
Find the lengths of the sides of the triangle.
The problem involves finding the side lengths of an isosceles right triangle given its area. Let's proceed with the solution.
Since the triangle is an isosceles right triangle, the two legs are equal, and the area is provided by the formula: For this triangle, both the base and the height are .
The area is given as 12.5, so we set up the equation:
Multiply both sides by 2 to solve for :
Take the square root of both sides:
Solve for :
Therefore, the length of each leg of the triangle is cm.
For the hypotenuse , use the Pythagorean theorem , where :
Thus, the lengths of the sides of the triangle are , , and .
Therefore, the correct solution is .
In front of you is an isosceles right triangle.
The expressions listed next to the sides describe their length.
( x>-8 length measurements in cm).
Since the area of the triangle is 32.
Find the lengths of the sides of the triangle.
To solve this problem, we'll utilize the properties of an isosceles right triangle and the area formula:
Now, let's work through each step:
Step 1: Recognize both legs of the isosceles triangle are .
Step 2: Apply the area formula for right triangles:
We know . Therefore, the equation is:
Step 3: Simplify and solve for .
Given the constraint , we discard since it violates the condition. Therefore,
Recalculate , which states the leg is 8.
Step 4: Determine the hypotenuse.
Therefore, the side lengths of the triangle are . Match this to the choices given, which is option 3.
The lengths of the sides of the triangle are .