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 $x$.

$$The equation in the problem is:

$x^2+10x+50=-4x+1$First, we identify that the equation is quadratic (and this is because the quadratic term in it does not cancel out), therefore, we will simplify the equation by moving all terms to one side and combine thelike terms:

$x^2+10x+50=-4x+1 \\ x^2+10x+4x+50-1 =0 \\ x^2+14x+49 =0 \\$

We want to solve this equation using factorization.

First, we'll check if we can find a common factor, but this is not possible, since there is no multiplicative factor common to all three terms on the left side of the equation.

We can factor the expression on the left side using the quadratic factoring formula for a trinomial, however, we prefer to factor it using the trinomial factoring methodl:

Note that the coefficient of the quadratic term (the term with the second power) is 1, and therefore we can try to perform factoring according to the quick trinomial method:

But before we do this in the problem - let's remember the general rule for factoring with the quick trinomial method:

The rule states that for the algebraic quadratic expression:

$x^2+bx+c$We can find a factorization to the form of a product if we can find two numbers $m,\hspace{4pt}n$such that the conditions (conditions of the quick trinomial method) are met:

$\begin{cases} m\cdot n=c\\ m+n=b \end{cases}$If we can find two such numbers $m,\hspace{4pt}n$then we can factor the general expression mentioned above into the form of a product and present it as:

$x^2+bx+c \\ \downarrow\\ (x+m)(x+n)$which is its factored form (product factors) of the expression,

Let's return now to the equation in the problem that we received in the last stage after arranging it:

$x^2+14x+49 =0$Note that the coefficients from the general form we mentioned in the rule above:

$x^2+bx+c$are:$\begin{cases} c=49 \\ b=14 \end{cases}$Don't forget to consider the coefficient together with its sign.

Let's continue - we want to factor the expression on the left side into factors according to the quick trinomial method, above, so we'll look for a pair of numbers $m,\hspace{4pt}n$ that satisfy:

$\begin{cases} m\cdot n=49\\ m+n=14 \end{cases}$We'll try to identify this pair of numbers using our knowledge of the multiplication table, we'll start from the multiplication between the two required numbers $m,\hspace{4pt}n$ that is - from the first row of the pair of requirements we mentioned in the last stage:

$m\cdot n=49$We identify that their product needs to give a positive result, and therefore we can conclude that their signs are identical.

Next, we'll refer to the factors (integers) of the number 49, and from our knowledge of the multiplication table we can know that there are only two possibilities for such factors: 7 and 7, or 49 and 1, as we previously concluded that their signs must be identical, a quick check of the two possibilities for the second condition:

$m+n=14$ will lead to a quick conclusion that the only possibility for fulfilling both of the above conditions together is:

$7,\hspace{4pt}7$That is:

$m=7,\hspace{4pt}n=7$(It doesn't matter which one we call m and which one we call n)

It is satisfied that:

$\begin{cases} \underline{7}\cdot \underline{7}=49\\ \underline{7}+\underline{7}=14 \end{cases}$ From here - we understood what the numbers we are looking for are and therefore we can factor the expression on the left side of the equation in question and present it as a product:

$x^2+14x+49 \\ \downarrow\\ (x+7)(x+7)$

In other words, we performed:

$x^2+bx+c \\ \downarrow\\ (x+m)(x+n)$

If so we factored the quadratic expression on the left side of the equation into factors using factoring according to the quick trinomial method, and the equation is:

$x^2+14x+49=0 \\ \downarrow\\ (x+7)(x+7)=0\\ (x+7)^2=0\\$$$$$In the last stage we notice that the expression on the left side the term:

$(x+7)$

is multiplied by itself and therefore the expression can be written as a squared term:

$(x+7)^2$

Now that the expression on the left side has been factored into a product form (in this case not just a product but actually a power form) we will continue to the quick solution of the equation we received:

$(x+7)^2=0$

Let's pay attention to a simple fact, on the left side there is a term that is raised to the second power, and on the right side the number 0.

0 squared (to the second power) will give the result 0, so we get that the equation equivalent to this equation is the equation:

$x+7=0$(We could have solved algebraically and taken the square root of both sides of the equation, we'll discuss this in a note at the end)

We'll solve this equation by transferring the constant number to the other side and we'll get that the only solution is:

$x=-7$Let's summarize then the stages of solving the quadratic equation using the quick trinomial factoring method:

$x^2+14x+49=0 \\ \downarrow\\ (x+7)(x+7)=0\\ (x+7)^2=0\\ \downarrow\\ x+7=0\\ x=-7$Therefore, the correct answer is answer B.

Note:

We could have reached the final equation by taking the square root of both sides of the equation, however - taking a square root involves considering two possibilities: positive and negative (it's enough to consider this only on one side, as described in the calculation below), that is, we could have performed:

$(x+7)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{8pt}}\\ \downarrow\\ \sqrt{(x+7)^2}=\pm\sqrt{0} \\ x+7=\pm0\\ x+7=0$

On the left side, the root (which is a half power) and the second power canceled each other out, and on the right side the root of 0 is 0, and we considered two possibilities positive and negative (this is the plus-minus sign indicated) except that the sign (which is actually multiplication by one or minus one) does not affect 0 which remains 0 in both cases, and therefore we reached the same equation we reached by logic - in the solution above.

In a case where on the right side there's a number other than 0, we could solve only by taking the root and considering the two positive and negative possibilities which would then give two different possibilities for the solution.

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 $6$.