Factoring by Short Multiplication - Examples, Exercises and Solutions

Understanding Factoring by Short Multiplication

Complete explanation with examples

The formulas for shortened multiplication will help us convert expressions with terms that have among them signs of addition or subtraction into expressions whose terms are multiplied.
The formulas for contracted multiplication are:
a2b2=(ab)×(a+b)a^2-b^2=(a-b)\times (a+b)

a2+2ab+b2=(a+b)2a^2+2ab+b^2=(a+b)^2

a22ab+b2=(ab)2a^2-2ab+b^2=(a-b)^2

Detailed explanation

Practice Factoring by Short Multiplication

Test your knowledge with 1 quizzes

Find the value of the parameter x.

\( -9x+3x^2=0 \)

Examples with solutions for Factoring by Short Multiplication

Step-by-step solutions included
Exercise #1

Find the value of the parameter x.

x2+x=0 x^2+x=0

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Factor the equation x2+x=0 x^2 + x = 0 .
  • Step 2: Use the Zero-Product Property to solve for x x .

Now, let's work through each step:

Step 1: Start by factoring the left-hand side of the equation:
x2+x=x(x+1) x^2 + x = x(x + 1)

Step 2: Apply the Zero-Product Property:
Since x(x+1)=0 x(x + 1) = 0 , we have two possible equations:
1) x=0 x = 0
2) x+1=0 x + 1 = 0

For the second equation, solve for x x :
x+1=0 x + 1 = 0 implies x=1 x = -1

Therefore, the solutions to the equation are x=0 x = 0 and x=1 x = -1 .

Hence, the value of the parameter x x is x=0,x=1 x = 0, x = -1 .

Answer:

x=0,x=1 x=0,x=-1

Video Solution
Exercise #2

Find the value of the parameter x.

9x+3x2=0 -9x+3x^2=0

Step-by-Step Solution

To solve the equation 9x+3x2=0 -9x + 3x^2 = 0 , follow these steps:

Step 1: Factor the equation.

Observe that both terms in the equation share a common factor, 3x 3x . We can factor this out:

9x+3x2=3x(3+x)-9x + 3x^2 = 3x(-3 + x).

The factored equation is 3x(3+x)=0 3x(-3 + x) = 0 .

Step 2: Apply the zero product property.

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two equations to solve:

  • 3x=0 3x = 0
  • 3+x=0 -3 + x = 0

Step 3: Solve each equation.

  • For 3x=0 3x = 0 , divide both sides by 3 to solve for x x :
  • x=0 x = 0
  • For 3+x=0 -3 + x = 0 , add 3 to both sides to solve for x x :
  • x=3 x = 3

Therefore, the solutions to the equation 9x+3x2=0 -9x + 3x^2 = 0 are x=0 x = 0 and x=3 x = 3 .

Matching these solutions to the given choices, the correct answer is choice 3: x=0,x=3 x = 0, x = 3 .

Thus, the values of x x that satisfy the equation are x=0 x = 0 and x=3 x = 3 .

Answer:

x=0,x=3 x=0,x=3

Video Solution
Exercise #3

Find the value of the parameter x.

9x312x2=0 9x^3-12x^2=0

Step-by-Step Solution

To solve this problem, we will factor the given polynomial expression:

Step 1: Identify the greatest common factor (GCF) in the equation 9x312x2=0 9x^3 - 12x^2 = 0 . The GCF of the terms 9x3 9x^3 and 12x2 12x^2 is 3x2 3x^2 .

Step 2: Factor out the GCF from the polynomial:

9x312x2=3x2(3x4)=0 9x^3 - 12x^2 = 3x^2(3x - 4) = 0 .

Step 3: Apply the zero-product property. Set each factor equal to zero:

  • 3x2=0 3x^2 = 0
  • 3x4=0 3x - 4 = 0

Step 4: Solve each equation for x x :

For 3x2=0 3x^2 = 0 , divide by 3:

x2=0 x^2 = 0 x=0 x = 0 .

For 3x4=0 3x - 4 = 0 , add 4 to both sides and then divide by 3:

3x=4 3x = 4
x=43 x = \frac{4}{3} .

Thus, the solutions to the equation 9x312x2=0 9x^3 - 12x^2 = 0 are x=0 x = 0 and x=43 x = \frac{4}{3} .

Therefore, the correct answer is:

x=0,x=43 x=0, x=\frac{4}{3}

Answer:

x=0,x=43 x=0,x=\frac{4}{3}

Video Solution
Exercise #4

Find the value of the parameter x.

x26x+8=0 x^2-6x+8=0

Step-by-Step Solution

To solve this quadratic equation by factoring, follow these steps:

  • Step 1: Write the equation: x26x+8=0 x^2 - 6x + 8 = 0 .
  • Step 2: Find two numbers that multiply to +8 +8 (the constant term) and add up to 6 -6 (the coefficient of x x ).

These numbers are 2 -2 and 4 -4 , since (2)×(4)=8 (-2) \times (-4) = 8 and (2)+(4)=6 (-2) + (-4) = -6 .

  • Step 3: Rewrite the quadratic expression as (x2)(x4)=0 (x - 2)(x - 4) = 0 .
  • Step 4: Solve each factor separately:
    • x2=0x=2 x - 2 = 0 \Rightarrow x = 2
    • x4=0x=4 x - 4 = 0 \Rightarrow x = 4

Therefore, the solutions to the quadratic equation x26x+8=0 x^2 - 6x + 8 = 0 are x=2 x = 2 and x=4 x = 4 .

The correct choice for the solution is:

x=2,x=4 x=2,x=4 which corresponds to choice 4.

Answer:

x=2,x=4 x=2,x=4

Video Solution

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