Find The Roots Of The Equation:$\[ 2p^2 + 7p - 30 = 0 \\]

===========================================================

Introduction

---

In mathematics, 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. In this article, we will focus on finding the roots of a quadratic equation, specifically the equation $2p^2 + 7p - 30 = 0$. We will use the quadratic formula and other methods to find the roots of this equation.

The Quadratic Formula

---

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. 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. In our case, $a = 2$, $b = 7$, and $c = -30$. Plugging these values into the quadratic formula, we get:

$$p = \frac{-7 \pm \sqrt{7^2 - 4(2)(-30)}}{2(2)}$$

Simplifying the Quadratic Formula

---

To simplify the quadratic formula, we need to calculate the value of the expression inside the square root. This expression is given by:

$$b^2 - 4ac = 7^2 - 4(2)(-30)$$

Expanding and simplifying this expression, we get:

$$b^2 - 4ac = 49 + 240 = 289$$

Now, we can plug this value back into the quadratic formula:

$$p = \frac{-7 \pm \sqrt{289}}{4}$$

Simplifying the Square Root

---

The square root of 289 is 17, so we can simplify the quadratic formula as follows:

$$p = \frac{-7 \pm 17}{4}$$

Finding the Roots

---

Now that we have simplified the quadratic formula, we can find the roots of the equation by plugging in the values of $p$. We get two possible values for $p$:

$$p_1 = \frac{-7 + 17}{4} = \frac{10}{4} = 2.5$$

$$p_2 = \frac{-7 - 17}{4} = \frac{-24}{4} = -6$$

Conclusion

---

In this article, we have used the quadratic formula to find the roots of the equation $2p^2 + 7p - 30 = 0$. We have simplified the quadratic formula and found two possible values for $p$. These values are $p_1 = 2.5$ and $p_2 = -6$. We can verify these values by plugging them back into the original equation.

Verification

---

To verify the values of $p$, we can plug them back into the original equation:

$$2p^2 + 7p - 30 = 0$$

Plugging in $p_1 = 2.5$, we get:

$$2(2.5)^2 + 7(2.5) - 30 = 2(6.25) + 17.5 - 30 = 12.5 + 17.5 - 30 = 0$$

This shows that $p_1 = 2.5$ is a root of the equation.

Plugging in $p_2 = -6$, we get:

$$2(-6)^2 + 7(-6) - 30 = 2(36) - 42 - 30 = 72 - 72 = 0$$

This shows that $p_2 = -6$ is also a root of the equation.

Real-World Applications

---

The quadratic formula has many real-world applications. For example, it can be used to model the motion of objects under the influence of gravity. It can also be used to find the maximum or minimum value of a quadratic function.

Conclusion

---

In conclusion, the quadratic formula is a powerful tool for finding the roots of a quadratic equation. We have used the quadratic formula to find the roots of the equation $2p^2 + 7p - 30 = 0$. We have simplified the quadratic formula and found two possible values for $p$. These values are $p_1 = 2.5$ and $p_2 = -6$. We can verify these values by plugging them back into the original equation.

Final Thoughts

---

The quadratic formula is a fundamental concept in mathematics, and it has many real-world applications. It is an essential tool for anyone who wants to study mathematics or science. In this article, we have used the quadratic formula to find the roots of a quadratic equation. We have simplified the quadratic formula and found two possible values for $p$. These values are $p_1 = 2.5$ and $p_2 = -6$. We can verify these values by plugging them back into the original equation.

References

---

• [1] "Quadratic Formula" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Quadratic Equation" by Wolfram MathWorld. Retrieved 2023-02-20.

Further Reading

---

• [1] "Quadratic Equations" by Khan Academy. Retrieved 2023-02-20.
• [2] "Quadratic Formula" by Purplemath. Retrieved 2023-02-20.
  

===========================================================

Introduction

---

In our previous article, we discussed how to find the roots of a quadratic equation using the quadratic formula. However, we know that there are many more questions that our readers may have. In this article, we will answer some of the most frequently asked questions about quadratic equations.

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 find the roots of a quadratic equation?

---

A: To find the roots of a quadratic equation, you can use the quadratic formula:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

---

A: The quadratic formula is a powerful tool for finding the roots of a quadratic equation. It is given by:

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

Q: How do I simplify the quadratic formula?

---

A: To simplify the quadratic formula, you need to calculate the value of the expression inside the square root. This expression is given by:

$$b^2 - 4ac = 7^2 - 4(2)(-30)$$

Expanding and simplifying this expression, you get:

$$b^2 - 4ac = 49 + 240 = 289$$

Q: What is the square root of 289?

---

A: The square root of 289 is 17.

Q: How do I find the roots of the equation $2p^2 + 7p - 30 = 0$?

---

A: To find the roots of the equation $2p^2 + 7p - 30 = 0$, you can use the quadratic formula:

$$p = \frac{-7 \pm \sqrt{289}}{4}$$

Simplifying the quadratic formula, you get:

$$p = \frac{-7 \pm 17}{4}$$

Q: What are the roots of the equation $2p^2 + 7p - 30 = 0$?

---

A: The roots of the equation $2p^2 + 7p - 30 = 0$ are $p_1 = 2.5$ and $p_2 = -6$.

Q: How do I verify the roots of a quadratic equation?

---

A: To verify the roots of a quadratic equation, you can plug them back into the original equation. If the equation is true, then the roots are correct.

Q: What are some real-world applications of quadratic equations?

---

A: Quadratic equations have many real-world applications, such as modeling the motion of objects under the influence of gravity, finding the maximum or minimum value of a quadratic function, and solving problems in physics, engineering, and economics.

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

---

A: Yes, you can use the quadratic formula to solve any quadratic equation. However, you need to make sure that the equation is in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What are some common mistakes to avoid when using the quadratic formula?

---

A: Some common mistakes to avoid when using the quadratic formula include:

• Not simplifying the expression inside the square root
• Not calculating the square root correctly
• Not plugging the roots back into the original equation to verify them
• Not using the correct values for $a$, $b$, and $c$

Conclusion

---

In this article, we have answered some of the most frequently asked questions about quadratic equations. We hope that this article has been helpful in clarifying any doubts that you may have had about quadratic equations. If you have any further questions, please don't hesitate to ask.

References

---

• [1] "Quadratic Formula" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Quadratic Equation" by Wolfram MathWorld. Retrieved 2023-02-20.

Further Reading

---

• [1] "Quadratic Equations" by Khan Academy. Retrieved 2023-02-20.
• [2] "Quadratic Formula" by Purplemath. Retrieved 2023-02-20.