Examples with solutions for The Quadratic Formula: Identifying and defining elements

Exercise #1

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

Identifies a,b,c

5x2+6x8=0 5x^2+6x-8=0

Video Solution

Step-by-Step Solution

To identify the coefficients from the quadratic equation 5x2+6x8=0 5x^2 + 6x - 8 = 0 , follow these steps:

  • Standard Form of Quadratic Equation: Compare 5x2+6x85x^2 + 6x - 8 with the standard form ax2+bx+c=0ax^2 + bx + c = 0.
  • Identify aa, bb, and cc:
    • a=5 a = 5 : The coefficient of x2x^2.
    • b=6 b = 6 : The coefficient of xx.
    • c=8 c = -8 : The constant term (independent of xx).

Therefore, from the equation 5x2+6x8=0 5x^2 + 6x - 8 = 0 , the coefficients are identified as a=5 a=5 , b=6 b=6 , and c=8 c=-8 .

Comparing with choices, we find that choice 2 is correct: a=5 a=5 , b=6 b=6 , c=8 c=-8 .

Thus, the coefficients are identified as a=5 a=5 , b=6 b=6 , c=8 c=-8 .

Answer

a=5 a=5 b=6 b=6 c=8 c=-8

Exercise #2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


8x25x+9=0 -8x^2-5x+9=0

What are the components of the equation?

Video Solution

Step-by-Step Solution

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

  • Step 1: Identify the equation is given in the standard quadratic form: 8x25x+9=0-8x^2 - 5x + 9 = 0.
  • Step 2: Determine each component:
    • aa is the coefficient of x2x^2, which is 8-8.
    • bb is the coefficient of xx, which is 5-5.
    • cc is the constant term, which is 99.

Now, let's resolve this using the above plan:

Step 1: The equation is already in standard form: 8x25x+9=0-8x^2 - 5x + 9 = 0.

Step 2: Recognize that:

  • a=8a = -8 because it is the coefficient of x2x^2.
  • b=5b = -5 because it is the coefficient of xx.
  • c=9c = 9 because it is the constant term.

Therefore, the components of the equation are a=8 a = -8 , b=5 b = -5 , c=9 c = 9 .

Answer

a=8 a=-8 b=5 b=-5 c=9 c=9

Exercise #3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


x2+4x5=0 x^2+4x-5=0

What are the components of the equation?

Video Solution

Step-by-Step Solution

The quadratic equation we have is x2+4x5=0 x^2 + 4x - 5 = 0 .

We'll compare this with the general form of a quadratic equation: ax2+bx+c=0 ax^2 + bx + c = 0 .

1. Identify a a : The coefficient of x2 x^2 in the given equation is 1 1 . Therefore, a=1 a = 1 .

2. Identify b b : The coefficient of x x in the given equation is 4 4 . Therefore, b=4 b = 4 .

3. Identify c c : The constant term in the given equation is 5 -5 . Therefore, c=5 c = -5 .

Thus, the components of the equation are:

  • a=1 a = 1
  • b=4 b = 4
  • c=5 c = -5

The correct answer to this problem, matching choice id 3, is:

a=1 a=1 b=4 b=4 c=5 c=-5

Answer

a=1 a=1 b=4 b=4 c=5 c=-5

Exercise #4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


x22=0 -x^2-2=0

What are the components of the equation?

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the coefficients in the quadratic equation given by x22=0-x^2 - 2 = 0.

The standard form of a quadratic equation is:

ax2+bx+c=0 ax^2 + bx + c = 0

In the given equation, x22=0-x^2 - 2 = 0, we can write it as:

1x2+0x2=0 -1 \cdot x^2 + 0 \cdot x - 2 = 0

This corresponds to the standard form with:

  • Coefficient aa: The coefficient of x2x^2 is 1-1.
  • Coefficient bb: The coefficient of xx is 00, as there is no xx term.
  • Coefficient cc: The constant term is 2-2.

Upon examining the answer choices given, the correct choice must match these coefficients precisely.

The correct choice is:

a=1 a = -1 , b=0 b = 0 , c=2 c = -2

This matches Choice 3.

Answer

a=1 a=-1 b=0 b=0 c=2 c=-2

Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


x2+7x=0 x^2+7x=0

What are the components of the equation?

Video Solution

Step-by-Step Solution

Let's solve the problem step-by-step:

First, consider the given equation:

x2+7x=0 x^2 + 7x = 0

This equation is almost in the standard form of a quadratic equation:

ax2+bx+c=0 ax^2 + bx + c = 0

Where:

  • a a is the coefficient of x2 x^2
  • b b is the coefficient of x x
  • c c is the constant term (independent number)

Let's identify each of these components from the given equation:

  • For a a : The term with x2 x^2 is x2 x^2 . In this case, the coefficient is implicitly 1, so a=1 a = 1 .
  • For b b : The term with x x is 7x 7x . The coefficient of x x is 7, so b=7 b = 7 .
  • For c c : There is no independent constant term visible, so we assume c=0 c = 0 .

Thus, the components of the quadratic equation are:

a=1 a = 1 , b=7 b = 7 , c=0 c = 0

The correct choice from the provided options is : a=1 a=1 , b=7 b=7 , c=0 c=0

Answer

a=1 a=1 b=7 b=7 c=0 c=0

Exercise #6

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


10x2+5+20x=0 10x^2+5+20x=0

What are the components of the equation?

Video Solution

Step-by-Step Solution

To determine the components of the quadratic equation, follow these steps:

  • Step 1: Recognize the standard form of a quadratic equation, which is ax2+bx+c=0 ax^2 + bx + c = 0 .
  • Step 2: Compare the given equation 10x2+20x+5=0 10x^2 + 20x + 5 = 0 to the standard form.
  • Step 3: Identify the coefficients:
    - The term 10x2 10x^2 indicates that a=10 a = 10 .
    - The term 20x 20x indicates that b=20 b = 20 .
    - The term 5 5 is the constant term, so c=5 c = 5 .

Therefore, the components of the equation are:

a=10 a = 10 , b=20 b = 20 , c=5 c = 5 .

The correct answer among the choices provided is the one that correctly identifies these coefficients:

a=10 a=10 b=20 b=20 c=5 c=5

Therefore, the correct choice is Choice 4.

Answer

a=10 a=10 b=20 b=20 c=5 c=5

Exercise #7

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


56x2+12x=0 5-6x^2+12x=0

What are the components of the equation?

Video Solution

Step-by-Step Solution

Let's solve this problem step-by-step by identifying the coefficients of the quadratic equation:

First, examine the given equation:

56x2+12x=05 - 6x^2 + 12x = 0

To make it easier to identify the coefficients, we rewrite the equation in the standard quadratic form:

6x2+12x+5=0-6x^2 + 12x + 5 = 0

In this expression, we can now directly identify the coefficients:

  • The coefficient of x2 x^2 (quadratic term) is a=6 a = -6 .
  • The coefficient of x x (linear term) is b=12 b = 12 .
  • The constant term (independent number) is c=5 c = 5 .

Thus, the components of the quadratic equation are:

a=6 a = -6 , b=12 b = 12 , c=5 c = 5

By comparing these values to the multiple-choice options, we can determine that the correct choice is:

Choice 4: a=6 a = -6 , b=12 b = 12 , c=5 c = 5

Therefore, the final solution is:

a=6 a = -6 , b=12 b = 12 , c=5 c = 5 .

Answer

a=6 a=-6 b=12 b=12 c=5 c=5

Exercise #8

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term


What is the value of c c in the function y=x2+25x y=-x^2+25x ?

Video Solution

Step-by-Step Solution

Let's recall the general form of the quadratic function:

y=ax2+bx+c y=ax^2+bx+c The function given in the problem is:

y=x2+25x y=-x^2+25x c c is the free term (meaning the coefficient of the term with power 0),

In the function in the problem there is no free term,

Therefore, we can identify that:

c=0 c=0 Therefore, the correct answer is answer A.

Answer

c=0 c=0

Exercise #9

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value ofb b in this quadratic equation:

y=4x216 y=4x^2-16

Video Solution

Step-by-Step Solution

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

  • Identify the given information and the standard quadratic form
  • Compare the given equation to this standard form
  • Extract the coefficients a a , b b , and c c and find b b

Now, let's work through each step:
Step 1: The problem provides us with the equation y=4x216 y = 4x^2 - 16 . It's already in a form where we can identify the coefficients.
Step 2: Recall the standard form of a quadratic equation is ax2+bx+c ax^2 + bx + c . Compare this form to the equation y=4x216 y = 4x^2 - 16 .
Step 3: By comparison, the coefficient of x2 x^2 (which is a a ) is 4. There is no x x term explicitly present, implying that b=0 b = 0 . The constant c c is -16.
Therefore, after comparison and identification, it becomes clear that the value of b b in the equation is b=0 b = 0 .

Answer

b=0 b=0

Exercise #10

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of c c in this quadratic equation:

y=5+3x2 y=5+3x^2

Video Solution

Step-by-Step Solution

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

  • Step 1: Compare the given equation y=5+3x2 y = 5 + 3x^2 to the standard form ax2+bx+c ax^2 + bx + c .
  • Step 2: Identify the terms corresponding to a a , b b , and c c .

Now, let's work through each step:
Step 1: The given equation is y=5+3x2 y = 5 + 3x^2 . Rearranging it in the standard form, we have y=3x2+0x+5 y = 3x^2 + 0\cdot x + 5 .

Step 2: From this arrangement, it's clear that:
- a=3 a = 3 (the coefficient of x2 x^2 )
- b=0 b = 0 (there is no x x term, so its coefficient is 0)
- c=5 c = 5 (the constant term)

Therefore, the value of c c is  c=5\ c=5 .

Answer

c=5 c=5

Exercise #11

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number


what is the value of b b in the equation

y=3x2+10x y=3x^2+10-x

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the coefficient of x x in the given quadratic equation. The equation given is y=3x2+10x y = 3x^2 + 10 - x . Let’s rearrange this equation to match the standard form of a quadratic equation ax2+bx+c ax^2 + bx + c .

The given equation can be rewritten as:

y=3x2x+10 y = 3x^2 - x + 10

Here, we can identify the coefficients:

  • a=3 a = 3 (for x2 x^2 )
  • b=1 b = -1 (for x x )
  • c=10 c = 10 (the constant term)

Therefore, the value of b b , the coefficient of x x , is 1 -1 .

Answer

b=1 b=-1

Exercise #12

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number


what is the value of a a in the equation

y=3x10+5x2 y=3x-10+5x^2

Video Solution

Step-by-Step Solution

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

  • Step 1: Rewrite the given equation in standard quadratic form if necessary.
  • Step 2: Identify the term with x2 x^2 in the equation.
  • Step 3: Extract the coefficient of x2 x^2 as a a .

Now, let's work through each step:
Step 1: The provided equation is y=3x10+5x2 y = 3x - 10 + 5x^2 . Although it's not initially in standard form, observation shows that the x2 x^2 term is clearly present.
Step 2: Locate the x2 x^2 term: in our equation, this term is 5x2 5x^2 .
Step 3: The coefficient of x2 x^2 is 5 5 . Hence, a=5 a = 5 .

Therefore, the coefficient of x2 x^2 , or a a , is a=5 a = 5 .

Answer

a=5 a=5

Exercise #13

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of c c in this quadratic equation:

y=5x2+4x3 y=-5x^2+4x-3

Video Solution

Step-by-Step Solution

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

  • Identify the given quadratic equation.
  • Compare the given equation with the standard quadratic form ax2+bx+c ax^2 + bx + c .
  • Determine the value of c c by direct comparison.

Now, let's work through each step:
Step 1: The given quadratic equation is y=5x2+4x3 y = -5x^2 + 4x - 3 .
Step 2: The standard form of a quadratic equation is ax2+bx+c ax^2 + bx + c . We need to match the coefficients accordingly.
Step 3: By comparing the terms from the equation with the standard form, a a is the coefficient of x2 x^2 , b b is the coefficient of x x , and c c is the constant term or the independent number.

Therefore, from the equation y=5x2+4x3 y = -5x^2 + 4x - 3 :

  • a=5 a = -5
  • b=4 b = 4
  • c=3 c = -3

Thus, the value of c c in the quadratic equation is c=3 c = -3 .

Answer

c=3 c=-3

Exercise #14

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of b b in the equation

y=2x3x2+1 y=2x-3x^2+1

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the coefficients in the given quadratic equation:

  • The equation provided is y=2x3x2+1 y = 2x - 3x^2 + 1 .
  • The standard form of a quadratic equation is ax2+bx+c ax^2 + bx + c .
  • From the equation, identify:
    • The ax2 ax^2 term is 3x2 -3x^2 , indicating a=3 a = -3 .
    • The bx bx term is 2x 2x , indicating b=2 b = 2 .
    • The constant term c c is 1 1 .

Thus, the coefficient b b in the equation y=2x3x2+1 y = 2x - 3x^2 + 1 is b=2 b = 2 , which corresponds to choice 1.

Answer

b=2 b=2

Exercise #15

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of a a in the equation

y=x23x+1 y=-x^2-3x+1

Video Solution

Step-by-Step Solution

To determine the coefficient a a in the given quadratic equation y=x23x+1 y = -x^2 - 3x + 1 , follow these steps:

  • Step 1: Recognize the form of the quadratic equation as y=ax2+bx+c y = ax^2 + bx + c .
  • Step 2: Identify the x2 x^2 term in the equation y=x23x+1 y = -x^2 - 3x + 1 .
  • Step 3: Determine the coefficient of the x2 x^2 term, which is in front of x2 x^2 .

In the equation y=x23x+1 y = -x^2 - 3x + 1 , the term involving x2 x^2 is x2-x^2, where the coefficient a a is clearly 1-1.

Hence, the value of a a is a=1 a = -1 .

Answer

a=1 a=-1

Exercise #16

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of a a in this quadratic equation:

y=3x+30 y=3x+30

Video Solution

Step-by-Step Solution

The given equation is y=3x+30 y = 3x + 30 . This equation does not include a term involving x2 x^2 , meaning it is not a quadratic equation. A quadratic equation is typically of the form ax2+bx+c=0 ax^2 + bx + c = 0 and includes an x2 x^2 term.

Let's compare:

  • The general form of a quadratic equation: ax2+bx+c ax^2 + bx + c
  • Our equation: y=3x+30 y = 3x + 30

By observation, the given equation does not have an x2 x^2 term. Therefore, there can be no coefficient a a because it would need to be a coefficient of an x2 x^2 component that does not exist in this equation.

Therefore, this is not a quadratic equation.

The correct choice is: "That's not a quadratic equation."

Answer

That's not a quadratic equation

Exercise #17

What is the value of a a in the equation 4x2=4x+16 4x^2=-4x+16 ?

Video Solution

Step-by-Step Solution

To solve the problem, we need to express the given equation 4x2=4x+16 4x^2 = -4x + 16 in the standard quadratic form ax2+bx+c=0 ax^2 + bx + c = 0 .

Let's perform the necessary transformations:
Rearrange the equation:

  • Start with: 4x2=4x+16 4x^2 = -4x + 16 .
  • Move all terms to one side to set the equation to zero:
    4x2+4x16=0 4x^2 + 4x - 16 = 0 .

We now have the standard quadratic form:
ax2+bx+c=0 ax^2 + bx + c = 0

In the equation 4x2+4x16=0 4x^2 + 4x - 16 = 0 , the coefficient of x2 x^2 is 4.

Thus, the value of a a is 4\boxed{4}.

Answer

a=4 a=4