Write The Following Equation In Standard Form. Then Solve.${ 6q^2 + 2q = 5q^2 - 3q + 36 }$The Equation In Standard Form Is { \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 quadratic equation in standard form, which is a crucial step in understanding the properties and behavior of quadratic functions. We will also explore the process of rewriting the equation in standard form and solving for the unknown variable.

Rewriting the Equation in Standard Form

The given equation is:

6q2+2q=5q2−3q+36{ 6q^2 + 2q = 5q^2 - 3q + 36 }

To rewrite this equation in standard form, we need to move all the terms to one side of the equation. We can do this by subtracting 5q25q^2 from both sides and adding 3q3q to both sides.

6q2+2q−5q2+3q=36{ 6q^2 + 2q - 5q^2 + 3q = 36 }

Simplifying the equation, we get:

q2+5q=36{ q^2 + 5q = 36 }

Now, we need to move the constant term to the right-hand side by subtracting 36 from both sides.

q2+5q−36=0{ q^2 + 5q - 36 = 0 }

This is the equation in standard form.

Solving the Quadratic Equation

To solve the quadratic equation, we can use the factoring method or the quadratic formula. In this case, we will use the factoring method.

We need to find two numbers whose product is -36 and whose sum is 5. These numbers are 9 and -4, since 9×(−4)=−369 \times (-4) = -36 and 9+(−4)=59 + (-4) = 5.

We can rewrite the equation as:

(q+9)(q−4)=0{ (q + 9)(q - 4) = 0 }

This tells us that either (q+9)=0(q + 9) = 0 or (q−4)=0(q - 4) = 0.

Solving for qq, we get:

q+9=0  ⟹  q=−9{ q + 9 = 0 \implies q = -9 }

q−4=0  ⟹  q=4{ q - 4 = 0 \implies q = 4 }

Therefore, the solutions to the quadratic equation are q=−9q = -9 and q=4q = 4.

Conclusion

In this article, we have learned how to rewrite a quadratic equation in standard form and solve for the unknown variable. We have also explored the process of factoring and using the quadratic formula to solve quadratic equations. By following these steps, we can solve quadratic equations and gain a deeper understanding of the properties and behavior of quadratic functions.

Standard Form of a Quadratic Equation

A quadratic equation in standard form is written as:

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

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

Properties of Quadratic Equations

Quadratic equations have several important properties, including:

  • Zeroes: A quadratic equation has two zeroes, which are the solutions to the equation.
  • Axis of Symmetry: The axis of symmetry is the vertical line that passes through the midpoint of the two zeroes.
  • Vertex: The vertex is the point on the graph of the quadratic function that is farthest from the axis of symmetry.

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems and make predictions about future trends.

Common Mistakes to Avoid

When solving quadratic equations, there are several common mistakes to avoid, including:

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying and solving the equation.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the solutions back into the original equation.
  • Not using the correct method: Make sure to use the correct method for solving the equation, such as factoring or the quadratic formula.

Conclusion

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 comprehensive Q&A guide to help you understand and solve 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. It is written in the form:

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

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

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

  • Factoring: If the equation can be factored into the product of two binomials, you can solve it by setting each factor equal to zero.
  • Quadratic Formula: If the equation cannot be factored, you can use the quadratic formula:

x=−b±b2−4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is written as:

x=−b±b2−4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

where aa, bb, and cc are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

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

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves finding two binomials whose product is the original equation, while the quadratic formula involves using a mathematical formula to find the solutions.

Q: Can I use the quadratic formula if the equation cannot be factored?

A: Yes, you can use the quadratic formula even if the equation cannot be factored. The quadratic formula will provide the solutions to the equation, even if they are not integers.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation and simplify. If the equation is true, then the solution is correct.

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

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

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying and solving the equation.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the solutions back into the original equation.
  • Not using the correct method: Make sure to use the correct method for solving the equation, such as factoring or the quadratic formula.

Q: Can I use technology to solve quadratic equations?

A: Yes, you can use technology to solve quadratic equations. Many calculators and computer software programs have built-in functions for solving quadratic equations.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations using factoring and the quadratic formula. Remember to check your solutions and avoid common mistakes. With practice and patience, you can become proficient in solving quadratic equations.