Difference of squares - Examples, Exercises and Solutions

We use the abbreviated multiplication formula:

$4-x^2=0$

We isolate the terms and extract the root:

$4=x^2$

$x=\sqrt{4}$

$x=\pm2$

To solve the equation $(\sqrt{x} + \frac{1}{2})(\sqrt{x} - \frac{1}{2}) = 0$, we can apply the zero-product property, which tells us that if a product of two factors is zero, at least one of the factors must be zero.

Let us proceed with each factor:

• First Factor: $\sqrt{x} + \frac{1}{2} = 0$
  Solving for $x$, subtract $\frac{1}{2}$ from both sides:
  $\sqrt{x} = -\frac{1}{2}$
  Squaring both sides, we get:
  $x = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}$.
  However, since the square root should be zero or positive, this case does not yield a real solution.
• Second Factor: $\sqrt{x} - \frac{1}{2} = 0$
  Solving for $x$, add $\frac{1}{2}$ to both sides:
  $\sqrt{x} = \frac{1}{2}$
  Squaring both sides, we have:
  $x = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.

Therefore, the solution to the equation $(\sqrt{x} + \frac{1}{2})(\sqrt{x} - \frac{1}{2}) = 0$ is $x = \frac{1}{4}$.

Upon reviewing the provided choices, the correct answer that matches our solution is: $\frac{1}{4}$ (Option 2).

To solve the equation $(x+3)(x-3) + (x+1)(x-1) = 0$, we will employ the difference of squares formula.

Step 1: Simplify $(x+3)(x-3)$ using the difference of squares:
$(x+3)(x-3) = x^2 - 3^2 = x^2 - 9$.

Step 2: Simplify $(x+1)(x-1)$ using the difference of squares:
$(x+1)(x-1) = x^2 - 1^2 = x^2 - 1$.

Step 3: Substitute the simplified expressions back into the original equation:
$x^2 - 9 + x^2 - 1 = 0$.

Step 4: Combine like terms:
$2x^2 - 10 = 0$.

Step 5: Simplify the equation by factoring or isolating $x^2$:
Divide through by 2 to get $x^2 - 5 = 0$.

Step 6: Solve for $x^2$:
$x^2 = 5$.

Step 7: Solve for $x$ by taking the square root of both sides:
$x = \pm \sqrt{5}$.

Therefore, the solution to the equation is $x = \pm \sqrt{5}$.

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

• Identify the given expression as a difference of squares.
• Apply the formula for finding the missing term in $(x+a)(x-a) = x^2 - a^2$.
• Determine the values of $a$ to fill in the blanks.

Now, let's work through each step:
Step 1: The expression given is $(x+\_—)\cdot(x-\_—) = x^2-121$. Recognize that $x^2 - 121$ is a difference of squares.
Step 2: We know from the difference of squares formula that $a^2 = 121$.
Step 3: Solve for $a$ by taking the square root of both sides: $a = \sqrt{121} = 11$.

This means the expression becomes: $(x+11)(x-11) = x^2 - 121$.

Therefore, the missing element is $11$.

To solve this problem, let's use the difference of squares formula, which is $(a + b)(a - b) = a^2 - b^2$. Given the equation $(_ + 3)(_- 3) = x^2 - 9$, we can compare it to the formula:

• $a^2 = x^2$ implies $a = x$.
• $b^2 = 9$ implies $b = 3$.

This means the expression $(_ + 3)(_- 3)$ should represent $(x + 3)(x - 3)$, satisfying the equation through the difference of squares formula.

Thus, the missing element to obtain a correct expression is $x$.

To solve the problem, begin with simplifying the left-hand side of the equation:

$(x+7)(x-7) = x^2 - 49$.

Thus, the original equation $\frac{(x+7)(x-7)}{3} = -11 - x^2$ simplifies to:

$\frac{x^2 - 49}{3} = -11 - x^2$.

Multiplying every term by 3 to clear the fraction, we obtain:

$x^2 - 49 = -33 - 3x^2$.

Add $3x^2$ to both sides to consolidate $x^2$ terms on one side:

$x^2 + 3x^2 = -33 + 49$.

This simplifies to:

$4x^2 = 16$.

Divide by 4 on both sides:

$x^2 = 4$.

Taking the square root of both sides provides:

$x = \pm 2$.

Therefore, the solution to the problem is $x = \pm 2$, corresponding to the choice labeled

±2

To solve the equation $\frac{2x^2-32}{8} = \frac{x+4}{2}$, let's proceed through these steps:

• Step 1: Simplify and eliminate fractions by cross-multiplying.
• Step 2: Rearrange and simplify the resulting equation.
• Step 3: Solve for the variable $x$.

Now, let's work through each step:

Step 1: Cross-multiply to eliminate the fractions.
We perform cross multiplication as follows:

This gives us:

Step 2: Rearrange and simplify the equation.
Move all terms to one side to set the equation to zero:

Simplify by dividing the entire equation by 4:

Step 3: Solve the quadratic equation by factoring.
Factor the quadratic equation:

Set each factor to zero and solve for $x$:

• $x - 6 = 0 \Rightarrow x = 6$
• $x + 4 = 0 \Rightarrow x = -4$

Considering the given multiple-choice answers, the correct solution is:

$x = 6$

Therefore, the solution to the problem is $x = 6$.

First we rearrange the equation and set it to 0

$4x^2-3-6=0$

$4x^2-9=0$

We then apply the shortcut multiplication formula:

$4(x^2-\frac{9}{4})=0$

$x^2-(\frac{3}{2})^2=0$

$(x-\frac{3}{2})(x+\frac{3}{2})=0$

$x=\pm\frac{3}{2}$

To solve the equation $(x+8)(8-x)+4(x-3)(x+3)+5(6-x^2)=0$, we will follow these steps:

• Step 1: Expand and simplify each factor using important algebraic formulas.
• Step 2: Combine all terms to form a quadratic equation.
• Step 3: Solve the quadratic equation using the quadratic formula.

Let's work through each step:

Step 1: Expand each part of the equation:

• The first term $(x+8)(8-x)$ is a difference of squares, which simplifies to:
  $(x+8)(8-x) = (8^2 - x^2) = 64 - x^2$.
• The second term $4(x-3)(x+3)$ is another difference of squares:
  $4[(x^2 - 9)] = 4x^2 - 36$.
• The third term $5(6-x^2)$ simplifies to:
  $30 - 5x^2$.

Step 2: Combine the results to form a quadratic equation:

Combine terms in the equation:

$64 - x^2 + 4x^2 - 36 + 30 - 5x^2 = 0$

Simplify further:

$(4x^2 - x^2 - 5x^2) + (64 - 36 + 30) = 0$

$-2x^2 + 58 = 0$

Rearrange to standard quadratic form:

$2x^2 = 58$

Step 3: Solve using the quadratic formula:

The equation simplifies to $x^2 = 29$.

Taking the square root of both sides gives the solutions:

$x = \pm \sqrt{29}$.

Thus, the solution to the equation is $\pm \sqrt{29}$.

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

• Step 1: Simplify the equation using the difference of squares.
• Step 2: Rearrange to form a suitable quadratic equation.
• Step 3: Solve the quadratic equation to find the value of $x$.

Now, let's work through each step:

Step 1: Simplify the left side using the difference of squares formula:

$(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7$

Step 2: Set the expressions equal and form a quadratic:

$x^2 - 7 = x^2 + 7x + 7$

Subtract $x^2$ from both sides:

$-7 = 7x + 7$

Rearrange the equation to isolate $x$:

$7x + 7 = -7$

Subtract 7 from both sides:

$7x = -14$

Divide both sides by 7:

$x = -2$

Therefore, the solution to the equation is $x = -2$.

To solve this problem, we need to recognize the expression $x^2 - 64$ as a difference of squares.

The difference of squares formula states: $a^2 - b^2 = (a - b)(a + b)$.

In this problem, we identify that:

• $a = x$

• $b^2 = 64$, which means $b = \sqrt{64} = 8$

Therefore, applying the formula gives us:

$x^2 - 64 = (x - 8)(x + 8)$

This indicates that the missing element in the expression $(x - \_)(\_ + x)$ is $8$.

Thus, the correct answer to fill in the missing element is $\boxed{8}$, corresponding to choice 4.

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

• Step 1: Recognize that $49 = 7^2$.
• Step 2: Apply the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$.
• Step 3: Compare the equation to the form and determine the blanks as $7$.

Now, let's work through each step:

Step 1: The given expression is $x^2 - 49$. Observe that 49 is a perfect square, written as $7^2$.

Step 2: According to the difference of squares formula, $x^2 - 49$ can be rewritten as $x^2 - 7^2$, which equals $(x - 7)(x + 7)$.

Step 3: Plugging in our values, we know the expression matches the form $(x - \_)\cdot(x + \_)$, with $7$ being the missing number.

Therefore, the solution to the problem is 7, which corresponds to choice 2.

To solve the problem, we need to express $x^2 - 6$ in the form of $(x-a)\cdot(x+a)$ because this represents the difference of squares, which is expressed as $(a-b)(a+b) = a^2 - b^2$.

We are given $x^2 - 6$. Compare this to the formula $x^2 - b^2$, it suggests that $b^2 = 6$.

The next step is to solve for $b$ by taking the square root of both sides:

$b^2 = 6 \Rightarrow b = \sqrt{6}$.

Thus, the missing number that completes the expression is $\sqrt{6}$.

Therefore, the solution to the problem is $\sqrt{6}$.

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

• Step 1: Identify the given expression and recognize the form.
• Step 2: Apply the difference of squares formula.
• Step 3: Determine the missing element.
• Step 4: Verify the solution against the possible choices.

Now, let's work through each step:
Step 1: The given expression is $x^2 - 36$. This resembles a difference of squares, which is $a^2 - b^2$.
Step 2: Recognize that $x^2$ represents $a^2$ and $36$ represents $b^2$.
Step 3: Find $b$ such that $b^2 = 36$. This gives $b = 6$ because $6^2 = 36$.
Step 4: The difference of squares formula states $a^2 - b^2 = (a - b)(a + b)$. So we rewrite $x^2 - 36$ as $(x - 6)(x + 6)$.

Therefore, the missing element that makes the expression true is $6$.

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

• Step 1: Expand the expression on the right side of the equation.
• Step 2: Compare it with the left-hand side equation and find the missing number.

Now, let's work through each step:
Step 1: Expand the expression $2(x-4)(x+4)$. Using the difference of squares, this becomes:

Step 2: Compare it with the original left side $2x^2 - \_ = 2x^2 - 32$.

The missing number must be 32 so that both sides of the equation are equal.

Therefore, the solution to the problem is $\textbf{32}$.