Solve The Equation: { (x+6)(x-2) = -7$}$

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Introduction

In this article, we will delve into the world of algebra and focus on solving a quadratic equation. The given equation is a product of two binomials, and our goal is to find the value of x that satisfies the equation. We will use various techniques and methods to solve this equation and provide a step-by-step solution.

Understanding the Equation

The given equation is (x+6)(x-2) = -7. This is a quadratic equation in the form of a product of two binomials. To solve this equation, we need to expand the product and then simplify the resulting expression.

Expanding the Product

To expand the product, we need to multiply the two binomials. We can do this by multiplying each term in the first binomial by each term in the second binomial.

(x+6)(x-2) = x(x) + x(-2) + 6(x) - 6(2)

Expanding the product, we get:

x^2 - 2x + 6x - 12

Combining like terms, we get:

x^2 + 4x - 12

Simplifying the Equation

Now that we have expanded the product, we can simplify the equation by setting it equal to -7.

x^2 + 4x - 12 = -7

To simplify the equation, we can add 7 to both sides of the equation.

x^2 + 4x - 12 + 7 = -7 + 7

This simplifies to:

x^2 + 4x - 5 = 0

Solving the Quadratic Equation

Now that we have simplified the equation, we can solve for x. We can use various methods to solve this quadratic equation, including factoring, the quadratic formula, and completing the square.

Factoring

One method to solve this equation is to factor the quadratic expression. We can look for two numbers whose product is -5 and whose sum is 4. These numbers are 5 and -1.

x^2 + 4x - 5 = (x + 5)(x - 1) = 0

Setting each factor equal to 0, we get:

x + 5 = 0 or x - 1 = 0

Solving for x, we get:

x = -5 or x = 1

Quadratic Formula

Another method to solve this equation is to use the quadratic formula. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 4, and c = -5. Plugging these values into the formula, we get:

x = (-(4) ± √((4)^2 - 4(1)(-5))) / 2(1)

Simplifying the expression, we get:

x = (-4 ± √(16 + 20)) / 2

x = (-4 ± √36) / 2

x = (-4 ± 6) / 2

Solving for x, we get:

x = (-4 + 6) / 2 or x = (-4 - 6) / 2

x = 2 / 2 or x = -10 / 2

x = 1 or x = -5

Completing the Square

Another method to solve this equation is to complete the square. We can rewrite the quadratic expression as:

x^2 + 4x = -5

Adding (4/2)^2 = 4 to both sides of the equation, we get:

x^2 + 4x + 4 = -5 + 4

This simplifies to:

(x + 2)^2 = -1

Taking the square root of both sides of the equation, we get:

x + 2 = ±√(-1)

x + 2 = ±i

Solving for x, we get:

x = -2 ± i

Conclusion

In this article, we have solved the quadratic equation (x+6)(x-2) = -7. We have used various methods to solve this equation, including factoring, the quadratic formula, and completing the square. We have found that the solutions to this equation are x = -5 and x = 1.

Final Answer

The final answer is x = -5 or x = 1.

Introduction

In our previous article, we solved the quadratic equation (x+6)(x-2) = -7 using various methods, including factoring, the quadratic formula, and completing the square. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

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

A: Factoring is a method of solving quadratic equations by expressing the quadratic expression as a product of two binomials. The quadratic formula, on the other hand, is a general method of solving quadratic equations that involves using the coefficients of the quadratic expression to find the solutions.

Q: Why do we need to add 7 to both sides of the equation when simplifying?

A: We need to add 7 to both sides of the equation to get rid of the negative term on the right-hand side. This allows us to simplify the equation and make it easier to solve.

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

A: Yes, the quadratic formula can be used to solve any quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the significance of the ± symbol in the quadratic formula?

A: The ± symbol in the quadratic formula indicates that there are two possible solutions to the equation. The ± symbol is used to represent the two possible values of x that satisfy the equation.

Q: Can we use completing the square to solve any quadratic equation?

A: Yes, completing the square can be used to solve any quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the difference between the quadratic formula and completing the square?

A: The quadratic formula is a general method of solving quadratic equations that involves using the coefficients of the quadratic expression to find the solutions. Completing the square, on the other hand, is a method of solving quadratic equations that involves rewriting the quadratic expression in a perfect square form.

Q: Can we use factoring to solve any quadratic equation?

A: Yes, factoring can be used to solve any quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the significance of the i symbol in the completing the square method?

A: The i symbol in the completing the square method represents the imaginary unit, which is a complex number that satisfies the equation i^2 = -1.

Q: Can we use the quadratic formula to solve equations with complex solutions?

A: Yes, the quadratic formula can be used to solve equations with complex solutions. In this case, the solutions will involve the imaginary unit i.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have about solving the quadratic equation (x+6)(x-2) = -7. We have covered various topics, including factoring, the quadratic formula, and completing the square.

Final Answer

The final answer is x = -5 or x = 1.

Additional Resources

  • For more information on solving quadratic equations, please refer to our previous article on the topic.
  • For more information on the quadratic formula, please refer to our article on the quadratic formula.
  • For more information on completing the square, please refer to our article on completing the square.

Related Articles

  • Solving Quadratic Equations: A Step-by-Step Guide
  • The Quadratic Formula: A General Method for Solving Quadratic Equations
  • Completing the Square: A Method for Solving Quadratic Equations

Tags

  • Quadratic equations
  • Factoring
  • Quadratic formula
  • Completing the square
  • Imaginary unit
  • Complex solutions