Solve For $t$.$d = -16t^2 + 4t$Choose The Correct Solution For $t$:A. $t = \frac{1}{2} \pm \frac{4 \sqrt{1 - 4d}}{2}$ B. $t = \frac{1}{8} \pm \frac{\sqrt{1 - 4d}}{8}$ C. $t = \frac{1}{2} \pm 4 \sqrt{1

by ADMIN 203 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of quadratic equation, which is in the form of d=−16t2+4td = -16t^2 + 4t. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Equation

The given equation is d=−16t2+4td = -16t^2 + 4t. This is a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where a=−16a = -16, b=4b = 4, and c=0c = 0. To solve for tt, we need to isolate tt on one side of the equation.

Step 1: Rearrange the Equation

The first step is to rearrange the equation to get all the terms on one side. We can do this by subtracting dd from both sides of the equation:

−16t2+4t−d=0-16t^2 + 4t - d = 0

Step 2: Use the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

In our case, a=−16a = -16, b=4b = 4, and c=−dc = -d. Plugging these values into the quadratic formula, we get:

t=−4±42−4(−16)(−d)2(−16)t = \frac{-4 \pm \sqrt{4^2 - 4(-16)(-d)}}{2(-16)}

Step 3: Simplify the Expression

Now, let's simplify the expression under the square root:

42−4(−16)(−d)=16−64d4^2 - 4(-16)(-d) = 16 - 64d

So, the expression becomes:

t=−4±16−64d−32t = \frac{-4 \pm \sqrt{16 - 64d}}{-32}

Step 4: Simplify Further

We can simplify the expression further by factoring out a −1-1 from the numerator and denominator:

t=4∓16−64d32t = \frac{4 \mp \sqrt{16 - 64d}}{32}

Now, let's simplify the expression under the square root:

16−64d=16(1−4d)16 - 64d = 16(1 - 4d)

So, the expression becomes:

t=4∓16(1−4d)32t = \frac{4 \mp \sqrt{16(1 - 4d)}}{32}

Step 5: Simplify the Square Root

We can simplify the square root by factoring out a 44:

16(1−4d)=42(1−4d)16(1 - 4d) = 4^2(1 - 4d)

So, the expression becomes:

t=4∓41−4d32t = \frac{4 \mp 4\sqrt{1 - 4d}}{32}

Step 6: Simplify Further

We can simplify the expression further by canceling out a factor of 44:

t=1∓1−4d8t = \frac{1 \mp \sqrt{1 - 4d}}{8}

Conclusion

In conclusion, the correct solution for tt is:

t=18±1−4d8t = \frac{1}{8} \pm \frac{\sqrt{1 - 4d}}{8}

This is option B in the given choices. Therefore, the correct answer is B.

Discussion

The solution to this problem involves several steps, including rearranging the equation, using the quadratic formula, and simplifying the expression. It's essential to follow each step carefully to ensure that the solution is correct.

In this article, we have provided a step-by-step guide to solving a quadratic equation. We have also discussed the importance of simplifying expressions and using the quadratic formula to solve quadratic equations.

Final Answer

The final answer is:

t = \frac{1}{8} \pm \frac{\sqrt{1 - 4d}}{8}$<br/> **Quadratic Equations: A Q&A Guide** ===================================== **Introduction** --------------- Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we provided a step-by-step guide to solving a quadratic equation in the form of $d = -16t^2 + 4t$. In this article, we will answer some frequently asked questions about quadratic equations. **Q: What is a quadratic equation?** ----------------------------------- A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. **Q: How do I solve a quadratic equation?** ----------------------------------------- To solve a quadratic equation, you can use the quadratic formula, which is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

You can also use factoring, completing the square, or graphing to solve quadratic equations.

Q: What is the quadratic formula?

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

Q: How do I use the quadratic formula?

To use the quadratic formula, you need to plug in the values of aa, bb, and cc into the formula. Then, simplify the expression under the square root and solve for xx.

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

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can I solve a quadratic equation by factoring?

Yes, you can solve a quadratic equation by factoring. To do this, you need to find two numbers whose product is acac and whose sum is bb. Then, you can write the quadratic equation as a product of two binomials.

Q: What is the relationship between the solutions of a quadratic equation and the discriminant?

The discriminant is the expression under the square root in the quadratic formula. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q: Can I solve a quadratic equation by graphing?

Yes, you can solve a quadratic equation by graphing. To do this, you need to graph the quadratic function and find the points where the graph intersects the x-axis. These points represent the solutions to the quadratic equation.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. We hope that this Q&A guide has provided you with a better understanding of quadratic equations and how to solve them.

Final Tips

  • Always read the problem carefully and understand what is being asked.
  • Use the quadratic formula or factoring to solve quadratic equations.
  • Check your solutions by plugging them back into the original equation.
  • Use graphing to visualize the solutions to quadratic equations.

Common Quadratic Equations

Here are some common quadratic equations:

  • x2+4x+4=0x^2 + 4x + 4 = 0
  • x2−6x+8=0x^2 - 6x + 8 = 0
  • x2+2x−6=0x^2 + 2x - 6 = 0

Solving Quadratic Equations: Practice Problems

Here are some practice problems to help you solve quadratic equations:

  • Solve the quadratic equation x2+2x−6=0x^2 + 2x - 6 = 0 using the quadratic formula.
  • Solve the quadratic equation x2−6x+8=0x^2 - 6x + 8 = 0 by factoring.
  • Solve the quadratic equation x2+4x+4=0x^2 + 4x + 4 = 0 by graphing.

We hope that this Q&A guide has provided you with a better understanding of quadratic equations and how to solve them.