This Equation Will Be In Standard Form If You Subtract $2x^2$ From Both Sides And Subtract $8x$ From Both Sides: 7 X 2 + 6 X + 8 = 2 X 2 − 8 X 7x^2 + 6x + 8 = 2x^2 - 8x 7 X 2 + 6 X + 8 = 2 X 2 − 8 X A. False B. True

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding their standard form is crucial for solving them. In this article, we will explore the concept of standard form in quadratic equations and provide a step-by-step guide on how to convert an equation into its standard form.

What is Standard Form?

Standard form in quadratic equations refers to the form of the equation where the coefficient of the squared term is equal to 1, and the equation is written in the format of $ax^2 + bx + c = 0$. The standard form is essential for solving quadratic equations using various methods, such as factoring, completing the square, and the quadratic formula.

Converting an Equation to Standard Form

To convert an equation to its standard form, we need to follow a series of steps. Let's consider the given equation: $7x^2 + 6x + 8 = 2x^2 - 8x$. Our goal is to subtract $2x^2$ from both sides and subtract $8x$ from both sides to obtain the equation in standard form.

Step 1: Subtract $2x^2$ from Both Sides

To start, we need to subtract $2x^2$ from both sides of the equation. This will help us isolate the squared term on one side of the equation.

# Subtract 2x^2 from both sides
equation = "7x^2 + 6x + 8 = 2x^2 - 8x"
new_equation = equation.replace("2x^2", "-2x^2")
print(new_equation)

Step 2: Simplify the Equation

After subtracting $2x^2$ from both sides, we need to simplify the equation by combining like terms.

# Simplify the equation
new_equation = "5x^2 + 6x + 8 = -8x"
print(new_equation)

Step 3: Subtract $8x$ from Both Sides

Next, we need to subtract $8x$ from both sides of the equation to isolate the linear term on one side.

# Subtract 8x from both sides
new_equation = "5x^2 + 6x + 8 = -8x"
new_equation = new_equation.replace("-8x", "+8x")
print(new_equation)

Step 4: Simplify the Equation

After subtracting $8x$ from both sides, we need to simplify the equation by combining like terms.

# Simplify the equation
new_equation = "5x^2 + 14x + 8 = 0"
print(new_equation)

Conclusion

In conclusion, the given equation $7x^2 + 6x + 8 = 2x^2 - 8x$ will be in standard form if we subtract $2x^2$ from both sides and subtract $8x$ from both sides. The resulting equation is $5x^2 + 14x + 8 = 0$, which is in standard form.

Answer

The correct answer is B. True.

Discussion

This problem requires a deep understanding of quadratic equations and their standard form. The student needs to be able to manipulate the equation by subtracting terms and simplifying the resulting expression. This problem is an excellent example of how to convert an equation to its standard form, which is a crucial step in solving quadratic equations.

Related Topics

• Quadratic equations
• Standard form
• Factoring
• Completing the square
• Quadratic formula

Practice Problems

1. Convert the equation $3x^2 - 4x + 2 = x^2 + 2x - 3$ to its standard form.
2. Solve the equation $2x^2 + 5x - 3 = 0$ using the quadratic formula.
3. Factor the equation $x^2 + 4x + 4 = 0$.

Conclusion

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding them can be a challenging task. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this complex topic.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. The method you choose depends on the specific equation and your personal preference.

Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. For example, the equation $x^2 + 5x + 6 = 0$ can be factored as $(x + 3)(x + 2) = 0$.

Completing the Square

Completing the square involves manipulating the quadratic equation to express it in the form $(x + h)^2 + k = 0$, where $h$ and $k$ are constants. This method is useful when the equation cannot be factored easily.

Quadratic Formula

The quadratic formula is a general method for solving quadratic equations. It states that the solutions to the equation $ax^2 + bx + c = 0$ are given by:

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

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants, and $a$ cannot be zero.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic equation is a parabola, which can intersect the x-axis at most two times.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It provides a general method for finding the solutions to any quadratic equation, regardless of whether it can be factored or not.

Q: Can I use the quadratic formula to solve a cubic equation?

A: No, the quadratic formula is only applicable to quadratic equations, which are polynomial equations of degree two. Cubic equations, which are polynomial equations of degree three, require a different method for solution.

Conclusion

In this article, we addressed some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this complex topic. Whether you are a student, teacher, or simply someone interested in mathematics, we hope this article has been helpful in clarifying your understanding of quadratic equations.

Related Topics

• Quadratic equations
• Factoring
• Completing the square
• Quadratic formula
• Linear equations
• Cubic equations

Practice Problems

1. Solve the equation $x^2 + 4x + 4 = 0$ using the quadratic formula.
2. Factor the equation $x^2 + 5x + 6 = 0$.
3. Determine the number of solutions to the equation $x^2 - 4x + 4 = 0$.

Glossary

• Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable (usually x) is two.
• Factoring: Expressing a quadratic equation as a product of two binomials.
• Completing the square: Manipulating a quadratic equation to express it in the form $(x + h)^2 + k = 0$, where $h$ and $k$ are constants.
• Quadratic formula: A general method for solving quadratic equations, which states that the solutions to the equation $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Linear equation: A polynomial equation of degree one, which means the highest power of the variable (usually x) is one.
• Cubic equation: A polynomial equation of degree three, which requires a different method for solution.