Solve The Quadratic Equation:$\[ -3x^2 + 5x - 7 = 0 \\]Write One Exact Solution In Each Box. You Can Add Or Remove Boxes. If There Are No Real Solutions, Remove All Boxes.$\[ \square \\] $\[ \square \\]

Mar 1, 2025 by ADMIN 206 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the quadratic equation $-3x^2 + 5x - 7 = 0$ . We will use the quadratic formula to find the exact solutions, and we will also discuss the importance of quadratic equations in real-world applications.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (in this case, $x$ ) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

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. The quadratic formula can be used to find the exact solutions of a quadratic equation, and it is a fundamental concept in algebra.

Solving the Quadratic Equation

Now that we have discussed the quadratic formula, let's apply it to the given quadratic equation $-3x^2 + 5x - 7 = 0$ . We can plug in the values of $a$ , $b$ , and $c$ into the quadratic formula to find the exact solutions.

x = \frac{-5 \pm \sqrt{5^2 - 4(-3)(-7)}}{2(-3)}

Simplifying the expression, we get:

x = \frac{-5 \pm \sqrt{25 - 84}}{-6}

x = \frac{-5 \pm \sqrt{-59}}{-6}

Since the square root of a negative number is not a real number, we can conclude that there are no real solutions to the quadratic equation.

Conclusion

In this article, we have discussed the quadratic formula and its application to solving quadratic equations. We have also seen that the quadratic equation $-3x^2 + 5x - 7 = 0$ has no real solutions. Quadratic equations are an important concept in mathematics, and they have numerous applications in real-world problems, such as physics, engineering, and economics.

Real-World Applications of Quadratic Equations

Quadratic equations have numerous applications in real-world problems. Some examples include:

Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity. For example, the trajectory of a projectile can be described using a quadratic equation.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings. For example, the stress on a beam can be described using a quadratic equation.
Economics: Quadratic equations are used to model economic systems, such as supply and demand curves. For example, the demand for a product can be described using a quadratic equation.

Tips and Tricks for Solving Quadratic Equations

Here are some tips and tricks for solving quadratic equations:

Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the exact solutions of a quadratic equation.
Check for real solutions: Before using the quadratic formula, check if the quadratic equation has real solutions. If the discriminant is negative, then the quadratic equation has no real solutions.
Simplify the expression: Simplify the expression under the square root before plugging in the values of $a$ , $b$ , and $c$ into the quadratic formula.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it has numerous applications in real-world problems. By following the tips and tricks outlined in this article, you can become proficient in solving quadratic equations and apply them to real-world problems.

Final Answer

Since the quadratic equation $-3x^2 + 5x - 7 = 0$ has no real solutions, we can conclude that there are no real solutions to the equation.

$\square$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will provide a Q&A guide to help you understand quadratic equations and how to solve them.

Q: What is a Quadratic Equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (in this case, $x$ ) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the Quadratic Formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions of a quadratic equation. It is given by:

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

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. You also need to simplify the expression under the square root before plugging in the values.

Q: What is the Discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$ . If the discriminant is negative, then the quadratic equation has no real solutions.

Q: How Do I Check for Real Solutions?

A: To check for real solutions, you need to check if the discriminant is negative. If the discriminant is negative, then the quadratic equation has no real solutions.

Q: What are Some Common Quadratic Equations?

A: Some common quadratic equations include:

$x^2 + 4x + 4 = 0$
$x^2 - 6x + 8 = 0$
$x^2 + 2x - 3 = 0$

Q: How Do I Factor a Quadratic Equation?

A: To factor a quadratic equation, you need to find two numbers whose product is $c$ and whose sum is $b$ . These numbers are the roots of the quadratic equation.

Q: What is the Difference Between a Quadratic Equation and a Linear Equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a parabolic shape, while a linear equation has a straight line shape.

Q: How Do I Use Quadratic Equations in Real-World Applications?

A: Quadratic equations have numerous applications in real-world problems, such as physics, engineering, and economics. For example, the trajectory of a projectile can be described using a quadratic equation, while the stress on a beam can be described using a quadratic equation.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the tips and tricks outlined in this article, you can become proficient in solving quadratic equations and apply them to real-world problems.

Final Answer

We hope that this Q&A guide has helped you understand quadratic equations and how to solve them. If you have any further questions, please don't hesitate to ask.

$\square$