Solve The Quadratic Equation By Using The Quadratic Formula:${ 25t^2 - 10t - 17 = 0 } F U L L Y S I M P L I F Y Y O U R A N S W E R , I N C L U D I N G A N Y N O N − R E A L S O L U T I O N S . Fully Simplify Your Answer, Including Any Non-real Solutions. F U Ll Ys Im Pl I F Yyo U R An S W Er , In C L U D In G An Y N O N − Re A L So L U T I O N S . { T = \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 delve into the world of quadratic equations and explore the quadratic formula, a powerful tool for solving these equations. We will use the quadratic formula to solve a specific quadratic equation, 25t2−10t−17=025t^2 - 10t - 17 = 0, and fully simplify our answer, including any non-real solutions.

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, tt) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants, and xx is the variable. 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=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the constants from the quadratic equation. The quadratic formula is a general solution, meaning that it will work for any quadratic equation, regardless of whether it can be factored or not.

Solving the Quadratic Equation

Now that we have the quadratic formula, let's use it to solve the quadratic equation 25t2−10t−17=025t^2 - 10t - 17 = 0. We can identify the constants aa, bb, and cc as follows:

  • a=25a = 25
  • b=−10b = -10
  • c=−17c = -17

Now, we can plug these values into the quadratic formula:

t=−(−10)±(−10)2−4(25)(−17)2(25)t = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(25)(-17)}}{2(25)}

Simplifying the expression, we get:

t=10±100+170050t = \frac{10 \pm \sqrt{100 + 1700}}{50}

t=10±180050t = \frac{10 \pm \sqrt{1800}}{50}

t=10±42.4350t = \frac{10 \pm 42.43}{50}

Now, we have two possible solutions:

t=10+42.4350t = \frac{10 + 42.43}{50}

t=52.4350t = \frac{52.43}{50}

t=1.049t = 1.049

t=10−42.4350t = \frac{10 - 42.43}{50}

t=−32.4350t = \frac{-32.43}{50}

t=−0.649t = -0.649

Fully Simplifying the Answer

We have found two possible solutions, t=1.049t = 1.049 and t=−0.649t = -0.649. However, we need to fully simplify our answer, including any non-real solutions. In this case, both solutions are real numbers, so we can stop here.

Conclusion

In this article, we used the quadratic formula to solve the quadratic equation 25t2−10t−17=025t^2 - 10t - 17 = 0. We identified the constants aa, bb, and cc, plugged them into the quadratic formula, and simplified the expression to find two possible solutions. We fully simplified our answer, including any non-real solutions, and found that both solutions are real numbers.

Real-World Applications

Quadratic equations have many 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.
  • Computer Science: Quadratic equations are used in computer graphics, game development, and machine learning.

Tips and Tricks

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 is a general solution, meaning that it will work for any quadratic equation, regardless of whether it can be factored or not.
  • Simplify the expression: Simplify the expression under the square root sign to make it easier to work with.
  • Check for non-real solutions: Check if the solutions are real numbers or complex numbers.
  • Use a calculator: Use a calculator to check your answers and simplify the expression.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

  • Not using the quadratic formula: Failing to use the quadratic formula can lead to incorrect solutions.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect solutions.
  • Not checking for non-real solutions: Failing to check for non-real solutions can lead to incorrect solutions.
  • Not using a calculator: Failing to use a calculator can lead to incorrect solutions.

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 is a general solution, meaning that it will work for any quadratic equation, regardless of whether it can be factored or not. By following the tips and tricks outlined in this article, you can avoid common mistakes and find the correct solutions to quadratic equations.

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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 answer some of the most frequently asked questions about quadratic equations, including how to solve them, what the quadratic formula is, and how to apply it to real-world problems.

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, tt) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants, and xx is the variable.

Q: What is the quadratic formula?

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

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the constants from the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the constants aa, bb, and cc from the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the steps to solve a quadratic equation?

A: The steps to solve a quadratic equation are:

  1. Identify the constants: Identify the constants aa, bb, and cc from the quadratic equation.
  2. Plug into the quadratic formula: Plug the values of aa, bb, and cc into the quadratic formula.
  3. Simplify the expression: Simplify the expression under the square root sign to make it easier to work with.
  4. Check for non-real solutions: Check if the solutions are real numbers or complex numbers.
  5. Use a calculator: Use a calculator to check your answers and simplify the expression.

Q: What are some common mistakes to avoid when solving quadratic equations?

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

  • Not using the quadratic formula: Failing to use the quadratic formula can lead to incorrect solutions.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect solutions.
  • Not checking for non-real solutions: Failing to check for non-real solutions can lead to incorrect solutions.
  • Not using a calculator: Failing to use a calculator can lead to incorrect solutions.

Q: How do I apply the quadratic formula to real-world problems?

A: The quadratic formula can be applied to a wide range of real-world problems, 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.
  • Computer Science: Quadratic equations are used in computer graphics, game development, and machine learning.

Q: What are some examples of quadratic equations in real-world problems?

A: Some examples of quadratic equations in real-world problems include:

  • Projectile motion: The trajectory of a projectile under the influence of gravity is a quadratic equation.
  • Spring-mass systems: The motion of a spring-mass system is a quadratic equation.
  • Electric circuits: The behavior of electric circuits is a quadratic equation.
  • Optimization problems: Quadratic equations are used to optimize systems, such as minimizing the cost of production or maximizing the profit.

Q: How do I check if a solution is real or complex?

A: To check if a solution is real or complex, you need to simplify the expression under the square root sign. If the expression is a perfect square, then the solution is real. If the expression is not a perfect square, then the solution is complex.

Q: What are some tips and tricks for solving quadratic equations?

A: Some tips and tricks for solving quadratic equations include:

  • Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
  • Simplify the expression: Simplify the expression under the square root sign to make it easier to work with.
  • Check for non-real solutions: Check if the solutions are real numbers or complex numbers.
  • Use a calculator: Use a calculator to check your answers and simplify the expression.

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 is a general solution, meaning that it will work for any quadratic equation, regardless of whether it can be factored or not. By following the steps outlined in this article, you can avoid common mistakes and find the correct solutions to quadratic equations.