The Quadratic Formula: Identifying and defining elements

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

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

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

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

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

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

• Step 1: Identify the equation is given in the standard quadratic form: $-8x^2 - 5x + 9 = 0$.
• Step 2: Determine each component:
  • $a$ is the coefficient of $x^2$, which is $-8$.
  • $b$ is the coefficient of $x$, which is $-5$.
  • $c$ is the constant term, which is $9$.

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

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

Step 2: Recognize that:

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

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

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

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

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

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

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

Thus, the components of the equation are:

• $a = 1$
• $b = 4$
• $c = -5$

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

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

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

The standard form of a quadratic equation is:

$ax^2 + bx + c = 0$

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

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

This corresponds to the standard form with:

• Coefficient $a$: The coefficient of $x^2$ is $-1$.
• Coefficient $b$: The coefficient of $x$ is $0$, as there is no $x$ term.
• Coefficient $c$: The constant term is $-2$.

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

The correct choice is:

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

This matches Choice 3.

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

First, consider the given equation:

$x^2 + 7x = 0$

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

$ax^2 + bx + c = 0$

Where:

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

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

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

Thus, the components of the quadratic equation are:

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

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

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

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

Therefore, the components of the equation are:

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

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

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

Therefore, the correct choice is Choice 4.

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

First, examine the given equation:

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

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

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

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

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

Thus, the components of the quadratic equation are:

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

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

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

Therefore, the final solution is:

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

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

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

$y=-x^2+25x$$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:

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

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 $a$, $b$, and $c$ and find $b$

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

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

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

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

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

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

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

The given equation can be rewritten as:

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

Here, we can identify the coefficients:

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

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

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 $x^2$ in the equation.
• Step 3: Extract the coefficient of $x^2$ as $a$.

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

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

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

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

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

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

• $a = -5$
• $b = 4$
• $c = -3$

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

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

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

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

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

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

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

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

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

Let's compare:

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

By observation, the given equation does not have an $x^2$ term. Therefore, there can be no coefficient $a$ because it would need to be a coefficient of an $x^2$ 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."

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

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

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

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

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

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