Solving Quadratic Equations using Factoring - Examples, Exercises and Solutions

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

• Step 1: Apply the distributive property to the equation.
• Step 2: Combine like terms.
• Step 3: Solve for $x$.

Now, let's work through each step:

Step 1: Distribute the number 5 across the expression inside the parentheses:
$3x + 5(x + 4) = 0$ becomes $3x + 5x + 20 = 0$.

Step 2: Combine the like terms:
Combine $3x$ and $5x$ to get $8x$.
Thus, the equation becomes $8x + 20 = 0$.

Step 3: Solve for $x$:
Subtract 20 from both sides: $8x = -20$.
Finally, divide both sides by 8: $x = \frac{-20}{8}$.

Simplify the fraction: $x = -2.5$.

Therefore, the solution to the equation is $x = -2.5$.

To solve the linear equation $8a + 2(3a - 7) = 0$, we'll proceed with the following steps:

Step 1: Apply the Distributive Property.
The equation given is $8a + 2(3a - 7) = 0$.
First, distribute the 2 across the terms inside the parenthesis:
$2(3a - 7) = 2 \times 3a + 2 \times (-7) = 6a - 14$.
By substituting this back into the equation, we have:
$8a + 6a - 14 = 0$.

Step 2: Combine Like Terms.
Now, combine the terms containing $a$:
$8a + 6a = 14a$.
The equation now becomes:
$14a - 14 = 0$.

Step 3: Isolate the Variable.
Add 14 to both sides of the equation to isolate terms with $a$:
$14a - 14 + 14 = 0 + 14$, which simplifies to:
$14a = 14$.
Next, divide both sides by 14 to solve for $a$:
$a = \frac{14}{14} = 1$.

Therefore, the solution to the equation is $a = 1$.

Let's proceed to solve the linear equation $3(a+1) - 3 = 0$:

Step 1: Distribute the 3 in the expression $3(a+1)$.

We get:
$3 \cdot a + 3 \cdot 1 - 3 = 0$

This simplifies to:
$3a + 3 - 3 = 0$

Step 2: Simplify the expression by combining like terms.

We simplify this to:
$3a + 0 = 0$ or simply $3a = 0$

Step 3: Isolate $a$ by dividing both sides by 3.

$\frac{3a}{3} = \frac{0}{3}$

Thus,
$a = 0$

Therefore, the solution to the problem is $a = 0$.

The correct choice is the option corresponding to $a = 0$.

To solve the given linear equation $5 - (3b - 1) = 0$, follow these steps:

• Step 1: Simplify the equation.
  Start by distributing the negative sign through the parentheses:
  $5 - 3b + 1 = 0$
• Step 2: Combine like terms.
  Combine the constant terms on the left side:
  $6 - 3b = 0$
• Step 3: Isolate the variable $b$.
  Subtract 6 from both sides of the equation to isolate the term with $b$:
  $-3b = -6$
• Step 4: Solve for $b$.
  Divide both sides by -3 to solve for $b$:
  $b = \frac{-6}{-3} = 2$

Therefore, the solution to the equation is $b = 2$.

Let's first expand the parentheses using the formula:

$a(x+b)=ax+ab$

$(2\times x)+(2\times4)+8=0$

$2x+8+8=0$

Next, we will substitute in our terms accordingly:

$2x+16=0$

Then, we will move the 16 to the left-hand side, keeping the appropriate sign:

$2x=-16$

Finally, we divide both sides by 2:

$\frac{2x}{2}=-\frac{16}{2}$

$x=-8$

To solve the equation $-2(-4 + y) - y = 0$, we will follow these steps:

• Step 1: Distribute $-2$ inside the parenthesis.
• Step 2: Simplify and combine like terms.
• Step 3: Solve the equation for $y$.

Let's proceed with the solution:

Step 1: Distribute $-2$ in the expression $-2(-4 + y)$. This will transform the expression as follows:

$-2(-4 + y) = -2 \times -4 + (-2) \times y = 8 - 2y$.

After distributing, the equation becomes:

$8 - 2y - y = 0$.

Step 2: Combine like terms. Notice that $-2y - y$ is equivalent to $-3y$:

$8 - 3y = 0$.

Step 3: Solve for $y$. First, isolate the term with $y$ by subtracting 8 from both sides:

$-3y = -8$.

Next, divide both sides by $-3$ to find $y$:

$y = \frac{-8}{-3} = \frac{8}{3}$.

Thus, the solution for $y$ is $\frac{8}{3}$, which can be written as a mixed number:

$y = 2\frac{2}{3}$.

Therefore, the solution to the problem is $y = 2\frac{2}{3}$.

We open the parentheses according to the formula:

$a(x+b)=ax+ab$

$5\times x+5\times3=0$

$5x+15=0$

We will move the 15 to the right section and keep the corresponding sign:

$5x=-15$

Divide both sections by 5

$\frac{5x}{5}=\frac{-15}{5}$

$x=-3$

To solve this equation, follow these steps:

• Step 1: Apply the distributive property to the equation:

  $2(4-x) = 2 \times 4 - 2 \times x = 8 - 2x$

• Step 2: Simplify the equation:

  The equation now becomes: $8 - 2x = 8$

• Step 3: Isolate the variable $x$ by simplifying the equation:

  First, subtract 8 from both sides:
  $8 - 2x - 8 = 8 - 8$
  This simplifies to:
  $-2x = 0$

• Step 4: Solve for $x$ by dividing both sides by -2:

  $x = \frac{0}{-2} = 0$

Therefore, the solution to the equation is $x = 0$.

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(7\times-2x)+(7\times5)=77$

We multiply accordingly

$-14x+35=77$

We will move the 35 to the right section and change the sign accordingly:

$-14x=77-35$

We solve the subtraction exercise on the right side and we will obtain:

$-14x=42$

We divide both sections by -14

$\frac{-14x}{-14}=\frac{42}{-14}$

$x=-3$

To open the parentheses on the left side, we'll use the formula:

$-a\left(b+c\right)=-ab-ac$

$-12a-24=27a$

We'll arrange the equation so that the terms with 'a' are on the right side, and maintain the plus and minus signs during the transfer:

$-24=27a+12a$

Let's group the terms on the right side:

$-24=39a$

Let's divide both sides by 39:

$-\frac{24}{39}=\frac{39a}{39}$

$-\frac{24}{39}=a$

Note that we can reduce the fraction since both numerator and denominator are divisible by 3:

$-\frac{8}{13}=a$

To solve the equation $6c + 7 + 4c = 3(c - 1)$, follow these steps:

• Step 1: Combine like terms on the left side of the equation.
  The like terms are $6c$ and $4c$. Combining these gives $10c + 7 = 3(c - 1)$.
• Step 2: Apply the distributive property on the right side of the equation.
  The term $3(c - 1)$ expands to $3c - 3$. Therefore, the equation becomes $10c + 7 = 3c - 3$.
• Step 3: Move all terms involving $c$ to one side and constants to the other.
  Subtract $3c$ from both sides: $10c - 3c + 7 = -3$ which simplifies to $7c + 7 = -3$.
• Step 4: Isolate the term with $c$ by subtracting 7 from both sides of the equation.
  This gives $7c = -3 - 7$ or $7c = -10$.
• Step 5: Solve for $c$.
  Divide both sides by 7: $c = \frac{-10}{7} = -\frac{10}{7}$. This can be converted to a mixed number, giving $-1\frac{3}{7}$.

Therefore, the solution to the equation is $c = -1\frac{3}{7}$. This corresponds to choice 2 in the provided answer choices.

To solve the equation $7y + 10y + 5 = 2(y + 3)$, let's proceed as follows:

• Step 1: Simplify the left side by combining like terms. The expression $7y + 10y$ combines to $17y$, so we have $17y + 5 = 2(y + 3)$.

• Step 2: Expand the right side. Distribute the 2 across the parenthesis: $2(y + 3)$ becomes $2y + 6$. The equation now reads $17y + 5 = 2y + 6$.

• Step 3: Isolate terms involving $y$ on one side. Subtract $2y$ from both sides: $17y - 2y + 5 = 6$, which simplifies to $15y + 5 = 6$.

• Step 4: Isolate $15y$ by subtracting 5 from both sides: $15y = 6 - 5$, which simplifies to $15y = 1$.

• Step 5: Solve for $y$ by dividing both sides by 15: $y = \frac{1}{15}$.

Therefore, the solution to the problem is $\mathbf{y = \frac{1}{15}}$.

To solve the linear equation $5(x-8)+\frac{1}{2}=0$, follow these steps:

• Step 1: Distribute the 5 across $(x-8)$.

The equation becomes:

$5x - 40 + \frac{1}{2} = 0$.

• Step 2: Combine the constants on the left side.

Combine $-40$ and $\frac{1}{2}$:

$5x - 40 + \frac{1}{2} = 0$.

Convert $-40$ into a fraction to simplify: $-40 = -\frac{80}{2}$.

The equation becomes:

$5x - \frac{80}{2} + \frac{1}{2} = 0$.

Simplify it to:

$5x - \frac{79}{2} = 0$.

• Step 3: Isolate $x$.

To move $-\frac{79}{2}$ to the other side, we add $\frac{79}{2}$ to both sides:

$5x = \frac{79}{2}$.

• Step 4: Solve for $x$.

Divide both sides by 5 to isolate $x$:

$x = \frac{79}{2} \div 5$.

$x = \frac{79}{2} \times \frac{1}{5}$.

$x = \frac{79}{10}$.

Therefore, the solution to the equation is $x = \frac{79}{10}$.

To solve the equation $-\frac{1}{2}(x+\frac{1}{4})=\frac{1}{8}$, we will first eliminate the fraction by multiplying both sides by the common denominator. The common denominator here is 8, so we proceed as follows:

• Step 1: Multiply both sides by 8 to eliminate the fractions:
  $8 \left(-\frac{1}{2}(x+\frac{1}{4})\right) = 8 \times \frac{1}{8}$
• Step 2: Simplify the left side:
  $-4(x+\frac{1}{4}) = 1$
• Step 3: Distribute $-4$ into the terms inside the parentheses:
  $-4x - 1 = 1$
• Step 4: Add 1 to both sides to isolate the term with $x$:
  $-4x = 2$
• Step 5: Divide both sides by $-4$ to solve for $x$:
  $x = \frac{2}{-4} = -\frac{1}{2}$

Therefore, the solution to the equation is $x = -\frac{1}{2}$.

We open the parentheses on the left side by the distributive property and use the formula:

$a(x+b)=ax+ab$

$-\frac{3}{2}x-12=\frac{1}{2}$

We multiply all terms by 2 to get rid of the fractions:

$-3x-12\times2=1$

$-3x-24=1$

We will move the minus 24 to the right section and keep the corresponding sign:

$-3x=24+1$

$-3x=25$

Divide both sections by minus 3:

$\frac{-3x}{-3}=\frac{25}{-3}$

$x=-\frac{25}{3}$