Quadratic Equations Practice Problems - Solve by 3 Methods

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 $a = 5$.

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 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.

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 $a = -1$.

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:

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