Solving Quadratic Equations using Factoring: One sided equations

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

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

$x=-3$

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 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 $x = 0$.

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 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 $b = 2$.

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 $a = 1$.

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

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 $x = -2.5$.

Let's open the parentheses using the distributive property and use the formula:

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

$40+5a-2a-14=56$

We'll substitute the terms accordingly:

$26+3a=56$

We'll move 26 to the right side and keep the appropriate sign:

$3a=56-26$

$3a=30$

We'll divide both sides by 3:

$\frac{3a}{3}=\frac{30}{3}$

$a=10$

To solve this equation, we follow these steps:

• Step 1: Distribute the coefficients within the parentheses.
• Step 2: Combine like terms.
• Step 3: Rearrange the equation to isolate and solve for $y$.

Let's apply these steps to solve the equation $3(y-2)+2(y+3)=180$:

Step 1: Distribute the terms:
$3(y-2) + 2(y+3) = 3y - 6 + 2y + 6$.

Step 2: Combine like terms:
Combine the $y$ terms and the constant terms: $3y + 2y - 6 + 6 = 5y$.
Thus, the equation simplifies to $5y = 180$.

Step 3: Solve for $y$:
To find $y$, divide both sides by 5:
$5y / 5 = 180 / 5$, which simplifies to $y = 36$.

Therefore, the solution to the equation is $y = 36$.

To solve the given equation $2(m+8) - 3(16+m) = 0$, we will follow these steps:

• Step 1: Apply the distributive property to expand the expression.
• Step 2: Combine like terms on both sides of the equation.
• Step 3: Solve for $m$ by isolating it.

Let's go through each step:

Step 1: Apply the distributive property:
Expand $2(m+8)$ to get $2m + 16$.
Expand $-3(16+m)$ to get $-48 - 3m$.

Step 2: Combine these expressions:
The equation becomes $2m + 16 - 48 - 3m = 0$.

Simplify it:
Combine like terms: $(2m - 3m) + (16 - 48)$.
This simplifies to $-m - 32 = 0$.

Step 3: Solve for $m$:
To isolate $m$, add 32 to both sides:
$-m = 32$.

Multiply both sides by -1 to solve for $m$:
$m = -32$.

Thus, the solution to the equation is $\mathbf{m = -32}$.

We will use the extended division rule and the formula:

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

$-42x+36+5+8x=0$

Let's input the appropriate terms:

$-34x+41=0$

We'll move -34x to the right side and maintain the appropriate sign:

$41=34x$

Let's divide both sides by 34:

$\frac{41}{34}=\frac{34x}{34}$

$\frac{41}{34}=x$

We'll convert the simple fraction to a mixed fraction:

$x=1\frac{7}{34}$

To solve the equation $7a(3+\frac{1}{a})-2a=0$, follow these steps:

• Step 1: Distribute and Simplify

  Begin by distributing the $7a$ across the terms inside the parentheses:

  $7a(3+\frac{1}{a}) = 7a \times 3 + 7a \times \frac{1}{a}$

  This simplifies to:

  $21a + 7$

  Thus, the equation becomes:

  $21a + 7 - 2a = 0$

• Step 2: Combine Like Terms

  Combine the 'a' terms on the left side:

  $21a - 2a + 7 = 0$

  This simplifies to:

  $19a + 7 = 0$

• Step 3: Solve for $a$

  Rearrange the equation to solve for $a$:

  $19a = -7$

  Divide both sides by 19 to isolate $a$:

  $a = -\frac{7}{19}$

Therefore, the solution to the equation is $a = -\frac{7}{19}$.

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

• Step 1: Apply the distributive property to expand the expression.
• Step 2: Simplify both sides to isolate the variable.
• Step 3: Divide to solve for the variable.

Now, let's work through each step:

Step 1: Apply the distributive property
The given equation is $-4(6-x)+(3x+5)=14$.
First, distribute $-4$:
$-4 \times 6 + -4 \times -x = -24 + 4x$.
Insert this into the equation:
$-24 + 4x + 3x + 5 = 14$.

Step 2: Simplify the equation
Combine like terms (the $x$ terms and constants):
$4x + 3x = 7x$.
$-24 + 5 = -19$.
So, the equation becomes:
$7x - 19 = 14$.

Step 3: Solve for $x$
Add 19 to both sides to isolate the $x$-term:
$7x - 19 + 19 = 14 + 19$.
$7x = 33$.
Divide both sides by 7:
$x = \frac{33}{7}$.
Simplifying this gives $x = 4\frac{5}{7}$.

Therefore, the solution to the problem is $x = 4\frac{5}{7}$.

To solve the linear equation $11(-3x+4)-7(6x-2)=3$, follow these steps:

Begin by applying the distributive property to the expression on both sides:

• Distribute $11$: $11(-3x+4) = 11 \cdot (-3x) + 11 \cdot 4 = -33x + 44$.
• Distribute $-7$: $-7(6x-2) = -7 \cdot 6x + -7 \cdot (-2) = -42x + 14$.

Substitute these results back into the equation:

$-33x + 44 - 42x + 14 = 3$

Combine like terms:

$(-33x - 42x) + (44 + 14) = 3$

$-75x + 58 = 3$

Isolate the term with $x$ by subtracting 58 from both sides:

$-75x = 3 - 58$

$-75x = -55$

Now, solve for $x$ by dividing both sides by $-75$:

$x = \frac{-55}{-75}$

Simplify the fraction by dividing both numerator and denominator by 5:

$x = \frac{11}{15}$

Therefore, the solution to the equation is $\boxed{\frac{11}{15}}$.