Given The Equation, Express It In Standard Form:$\[ Y = 2x^2 + 8x + 7 \\]

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is:

${ ax^2 + bx + c = 0 }$

where a, b, and c are constants, and x is the variable.

Expressing the Given Equation in Standard Form

The given equation is:

${ y = 2x^2 + 8x + 7 }$

To express this equation in standard form, we need to rewrite it in the form:

${ ax^2 + bx + c = 0 }$

where a, b, and c are constants.

Step 1: Identify the Coefficients

The given equation is already in the form:

${ y = 2x^2 + 8x + 7 }$

We can identify the coefficients as follows:

• a = 2 (coefficient of x^2)
• b = 8 (coefficient of x)
• c = 7 (constant term)

Step 2: Rewrite the Equation in Standard Form

Now that we have identified the coefficients, we can rewrite the equation in standard form:

${ 2x^2 + 8x + 7 = 0 }$

This is the standard form of the given equation.

Step 3: Factor the Equation (Optional)

If the equation can be factored, we can factor it to express it in a more simplified form. However, in this case, the equation cannot be factored, so we will leave it in standard form.

Step 4: Solve the Equation (Optional)

If we need to solve the equation, we can use various methods, such as factoring, quadratic formula, or graphing. However, in this case, we are only interested in expressing the equation in standard form.

Conclusion

In this article, we have expressed the given quadratic equation in standard form. We identified the coefficients, rewrote the equation in standard form, and left it in that form since it cannot be factored. This is a fundamental concept in mathematics, and it is essential to understand how to express quadratic equations in standard form.

Common Quadratic Equations

Here are some common quadratic equations that can be expressed in standard form:

• ${ x^2 + 4x + 4 = 0 }$
• ${ x^2 - 6x + 8 = 0 }$
• ${ x^2 + 2x - 6 = 0 }$

These equations can be expressed in standard form by identifying the coefficients and rewriting the equation in the form:

${ ax^2 + bx + c = 0 }$

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and cost-benefit analysis.

Conclusion

Frequently Asked Questions

Quadratic equations are a fundamental concept in mathematics, and they can be confusing at times. Here are some frequently asked questions about quadratic equations, along with their answers.

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 is two. The general form of a quadratic equation is:

${ ax^2 + bx + c = 0 }$

where a, b, and c are constants, and x is the variable.

Q: How do I identify the coefficients in a quadratic equation?

A: To identify the coefficients in a quadratic equation, you need to look at the terms of the equation. The coefficient of the x^2 term is the number in front of the x^2 term, the coefficient of the x term is the number in front of the x term, and the constant term is the number that is not multiplied by any variable.

Q: How do I rewrite a quadratic equation in standard form?

A: To rewrite a quadratic equation in standard form, you need to rearrange the terms of the equation so that the x^2 term is first, followed by the x term, and then the constant term. For example, if you have the equation:

${ 2x^2 + 8x + 7 = 3x^2 + 2x - 2 }$

You can rewrite it in standard form by subtracting 3x^2 from both sides, adding 2x to both sides, and adding 2 to both sides:

${ -x^2 + 6x + 9 = 0 }$

Q: Can I factor a quadratic equation?

A: Yes, you can factor a quadratic equation if it can be expressed as the product of two binomials. For example, the equation:

${ x^2 + 4x + 4 = 0 }$

Can be factored as:

${ (x + 2)(x + 2) = 0 }$

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, quadratic formula, and graphing. The method you choose depends on the type of equation and the information you need to find.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation of the form:

${ ax^2 + bx + c = 0 }$

The formula is:

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

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula:

${ b^2 - 4ac }$

If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can I use the quadratic formula to solve a quadratic equation that cannot be factored?

A: Yes, you can use the quadratic formula to solve a quadratic equation that cannot be factored. The quadratic formula will give you the solutions to the equation, even if it cannot be factored.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to find the x and y coordinates of the graph.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. The vertex can be found using the formula:

${ x = \frac{-b}{2a} }$

Q: How do I find the x-intercepts of a quadratic equation?

A: To find the x-intercepts of a quadratic equation, you can set the equation equal to zero and solve for x. The x-intercepts are the points where the graph crosses the x-axis.

Q: Can I use a quadratic equation to model real-world situations?

A: Yes, you can use a quadratic equation to model real-world situations, such as the motion of an object under the influence of gravity, the growth of a population, or the cost of a product.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding how to identify the coefficients, rewrite the equation in standard form, and solve the equation, you can use quadratic equations to model and analyze real-world situations.