Solve For $g$.$16g^2 - 15 = 21$Write Your Answers As Integers Or As Proper Fractions:$ G = □ G = \square G = □ [/tex] Or $g = \square$

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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 focus on solving a specific quadratic equation, $16g^2 - 15 = 21$, to find the value of $g$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $g$) 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 our equation, $16g^2 - 15 = 21$, we can rewrite it in the standard form as $16g^2 - 36 = 0$ by subtracting 21 from both sides.

Step 1: Rearrange the Equation

To solve the equation, we need to isolate the variable $g$. We can start by rearranging the equation to get all the terms involving $g$ on one side. In this case, we can add 36 to both sides of the equation to get:

16g2=3616g^2 = 36

Step 2: Divide Both Sides by 16

Next, we need to get rid of the coefficient of $g^2$, which is 16. We can do this by dividing both sides of the equation by 16:

g2=3616g^2 = \frac{36}{16}

Step 3: Simplify the Fraction

The fraction $\frac{36}{16}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

g2=94g^2 = \frac{9}{4}

Step 4: Take the Square Root

To find the value of $g$, we need to take the square root of both sides of the equation. Since we are looking for the value of $g$, we will take the positive square root:

g=94g = \sqrt{\frac{9}{4}}

Step 5: Simplify the Square Root

The square root of $\frac{9}{4}$ can be simplified by taking the square root of the numerator and the denominator separately:

g=94g = \frac{\sqrt{9}}{\sqrt{4}}

Step 6: Simplify the Square Roots

The square roots of 9 and 4 can be simplified as follows:

g=32g = \frac{3}{2}

Conclusion

In this article, we solved the quadratic equation $16g^2 - 15 = 21$ to find the value of $g$. We broke down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer. The value of $g$ is $\frac{3}{2}$.

Final Answer

g=32g = \frac{3}{2}

Discussion

This solution demonstrates the importance of following the order of operations and using algebraic manipulations to solve quadratic equations. It also highlights the need to simplify fractions and square roots to arrive at the final answer. If you have any questions or would like to discuss this solution further, please feel free to leave a comment below.

Related Topics

  • Quadratic equations
  • Algebraic manipulations
  • Simplifying fractions and square roots
  • Solving quadratic equations

References

  • [1] "Quadratic Equations" by Khan Academy
  • [2] "Algebraic Manipulations" by Math Open Reference
  • [3] "Simplifying Fractions and Square Roots" by Purplemath
    Quadratic Equation Q&A: Solving for $g$ =====================================================

Introduction

In our previous article, we solved the quadratic equation $16g^2 - 15 = 21$ to find the value of $g$. We received many questions and comments from readers, and we're excited to address them in this Q&A article. Whether you're a student struggling with quadratic equations or a professional looking for a refresher, this article is for you.

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 (in this case, $g$) 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 solve a quadratic equation?

A: To solve a quadratic equation, you need to follow these steps:

  1. Rearrange the equation to get all the terms involving the variable on one side.
  2. Divide both sides of the equation by the coefficient of the variable squared.
  3. Simplify the fraction and take the square root of both sides.
  4. Simplify the square root and arrive at the final answer.

Q: What if I have a quadratic equation with a negative coefficient?

A: If you have a quadratic equation with a negative coefficient, you can multiply both sides of the equation by -1 to get a positive coefficient. This will not change the solution to the equation.

Q: Can I use a calculator to solve quadratic equations?

A: Yes, you can use a calculator to solve quadratic equations. However, it's always a good idea to understand the steps involved in solving the equation and to check your answer using a calculator.

Q: What if I have a quadratic equation with a complex solution?

A: If you have a quadratic equation with a complex solution, you can use the quadratic formula to find the solution. The quadratic formula is:

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

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, it's always a good idea to check your answer using a calculator or by plugging the values back into the original equation.

Q: What if I have a quadratic equation with a repeated root?

A: If you have a quadratic equation with a repeated root, you can use the factored form of the equation to find the solution. The factored form of a quadratic equation is:

(x−r)2=0(x - r)^2 = 0

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

A: Yes, you can use the factored form to solve any quadratic equation. However, it's always a good idea to check your answer using a calculator or by plugging the values back into the original equation.

Conclusion

In this Q&A article, we addressed many common questions and concerns about solving quadratic equations. Whether you're a student struggling with quadratic equations or a professional looking for a refresher, we hope this article has been helpful. Remember to always follow the steps involved in solving a quadratic equation and to check your answer using a calculator or by plugging the values back into the original equation.

Final Answer

g=32g = \frac{3}{2}

Discussion

This Q&A article demonstrates the importance of understanding the steps involved in solving quadratic equations and the need to check your answer using a calculator or by plugging the values back into the original equation. If you have any questions or would like to discuss this article further, please feel free to leave a comment below.

Related Topics

  • Quadratic equations
  • Algebraic manipulations
  • Simplifying fractions and square roots
  • Solving quadratic equations
  • Quadratic formula
  • Factored form of a quadratic equation

References

  • [1] "Quadratic Equations" by Khan Academy
  • [2] "Algebraic Manipulations" by Math Open Reference
  • [3] "Simplifying Fractions and Square Roots" by Purplemath
  • [4] "Quadratic Formula" by Math Is Fun
  • [5] "Factored Form of a Quadratic Equation" by Mathway