Haley Substituted The Values Of \[$a, B,\$\] And \[$c\$\] Into The Quadratic Formula Below.$\[ X = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(7)(-2)}}{2(7)} \\]What Was Haley's Function In Standard Form?$\[ F(x) = 7x^2 + 10x + 2

Solving Quadratic Equations: A Step-by-Step Guide to Haley's Function

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore how to solve a quadratic equation using the quadratic formula, and we will apply this concept to a specific problem presented by Haley. We will also discuss the standard form of a quadratic function and how to identify it.

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation in the form $ax^2 + bx + c = 0$.

Haley was given a quadratic equation in the form $ax^2 + bx + c = 0$, and she was asked to substitute the values of $a$, $b$, and $c$ into the quadratic formula to find the solutions. The quadratic equation was:

${ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(7)(-2)}}{2(7)} }$

To solve this problem, Haley needed to substitute the values of $a$, $b$, and $c$ into the quadratic formula. Let's break down the problem step by step.

Step 1: Identify the values of $a$, $b$, and $c$

In the quadratic equation $x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(7)(-2)}}{2(7)}$, we can identify the values of $a$, $b$, and $c$ as follows:

• $a = 7$
• $b = 10$
• $c = -2$

Step 2: Substitute the values of $a$, $b$, and $c$ into the quadratic formula

Now that we have identified the values of $a$, $b$, and $c$, we can substitute them into the quadratic formula:

${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$

Substituting the values of $a$, $b$, and $c$, we get:

${ x = \frac{-10 \pm \sqrt{10^2 - 4(7)(-2)}}{2(7)} }$

Step 3: Simplify the expression

Now that we have substituted the values of $a$, $b$, and $c$ into the quadratic formula, we can simplify the expression:

${ x = \frac{-10 \pm \sqrt{100 + 56}}{14} }$

${ x = \frac{-10 \pm \sqrt{156}}{14} }$

${ x = \frac{-10 \pm 12.5}{14} }$

Step 4: Solve for $x$

Now that we have simplified the expression, we can solve for $x$:

${ x = \frac{-10 + 12.5}{14} }$

${ x = \frac{2.5}{14} }$

${ x = 0.18 }$

${ x = \frac{-10 - 12.5}{14} }$

${ x = \frac{-22.5}{14} }$

${ x = -1.61 }$

A quadratic function is a polynomial function of degree two, and it can be written in the form:

${ f(x) = ax^2 + bx + c }$

where $a$, $b$, and $c$ are constants.

To identify the standard form of a quadratic function, we need to look for the following characteristics:

• The function must be a polynomial of degree two.
• The function must have a leading coefficient of $a$.
• The function must have a linear term of $bx$.
• The function must have a constant term of $c$.

Haley's function is given by:

${ f(x) = 7x^2 + 10x + 2 }$

This function meets all the characteristics of a quadratic function in standard form:

• The function is a polynomial of degree two.
• The leading coefficient is $a = 7$.
• The linear term is $bx = 10x$.
• The constant term is $c = 2$.

Therefore, Haley's function is in standard form.

In this article, we have explored how to solve a quadratic equation using the quadratic formula, and we have applied this concept to a specific problem presented by Haley. We have also discussed the standard form of a quadratic function and how to identify it. By following these steps, students can master the skill of solving quadratic equations and identifying the standard form of a quadratic function.
Quadratic Equations: A Q&A Guide

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we explored how to solve a quadratic equation using the quadratic formula and applied this concept to a specific problem presented by Haley. In this article, we will provide a Q&A guide to help students better understand quadratic equations and how to solve them.

A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation in the form $ax^2 + bx + c = 0$.

A: To use the quadratic formula to solve a quadratic equation, you need to follow these steps:

1. Identify the values of $a$, $b$, and $c$ in the quadratic equation.
2. Substitute the values of $a$, $b$, and $c$ into the quadratic formula.
3. Simplify the expression.
4. Solve for $x$.

A: The standard form of a quadratic function is a polynomial function of degree two, and it can be written in the form:

${ f(x) = ax^2 + bx + c }$

where $a$, $b$, and $c$ are constants.

A: To identify the standard form of a quadratic function, you need to look for the following characteristics:

• The function must be a polynomial of degree two.
• The function must have a leading coefficient of $a$.
• The function must have a linear term of $bx$.
• The function must have a constant term of $c$.

A: There are three different types of solutions to a quadratic equation:

• Real and distinct solutions: These are solutions that are real numbers and are distinct from each other.
• Real and repeated solutions: These are solutions that are real numbers and are repeated.
• Complex solutions: These are solutions that are complex numbers.

A: To determine the type of solution to a quadratic equation, you need to look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is:

• Positive, the solutions are real and distinct.
• Zero, the solutions are real and repeated.
• Negative, the solutions are complex.

A: Some common mistakes to avoid when solving quadratic equations include:

• Not simplifying the expression: Make sure to simplify the expression before solving for $x$.
• Not checking for complex solutions: Make sure to check for complex solutions, especially if the discriminant is negative.
• Not using the correct formula: Make sure to use the correct formula for the quadratic equation, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

In this article, we have provided a Q&A guide to help students better understand quadratic equations and how to solve them. By following these steps and avoiding common mistakes, students can master the skill of solving quadratic equations and become more confident in their math abilities.