What Are The Two Possible Solutions To Each Of The Equations Below?8. $3(x-5)^2 = 48$9. $2x^2 - 56 = 42$

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What are the Two Possible Solutions to Each of the Equations Below?

In mathematics, solving equations is a crucial aspect of problem-solving. Equations can be linear or quadratic, and they can have one or more solutions. In this article, we will explore two possible solutions to each of the given equations. We will use algebraic methods to solve these equations and provide step-by-step solutions.

Equation 1: 3(x−5)2=483(x-5)^2 = 48

Step 1: Expand the Equation

To solve the equation 3(x−5)2=483(x-5)^2 = 48, we need to expand the squared term.

3(x-5)^2 = 48
3(x^2 - 10x + 25) = 48
3x^2 - 30x + 75 = 48

Step 2: Rearrange the Equation

Next, we need to rearrange the equation to set it equal to zero.

3x^2 - 30x + 75 - 48 = 0
3x^2 - 30x + 27 = 0

Step 3: Solve the Quadratic Equation

Now, we can use the quadratic formula to solve the equation.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x = \frac{-(-30) \pm \sqrt{(-30)^2 - 4(3)(27)}}{2(3)}
x = \frac{30 \pm \sqrt{900 - 324}}{6}
x = \frac{30 \pm \sqrt{576}}{6}
x = \frac{30 \pm 24}{6}

Step 4: Simplify the Solutions

Finally, we can simplify the solutions to get the two possible values of x.

x = \frac{30 + 24}{6} = \frac{54}{6} = 9
x = \frac{30 - 24}{6} = \frac{6}{6} = 1

Equation 2: 2x2−56=422x^2 - 56 = 42

Step 1: Rearrange the Equation

To solve the equation 2x2−56=422x^2 - 56 = 42, we need to rearrange it to set it equal to zero.

2x^2 - 56 - 42 = 0
2x^2 - 98 = 0

Step 2: Divide the Equation by 2

Next, we can divide the equation by 2 to simplify it.

x^2 - 49 = 0

Step 3: Factor the Equation

Now, we can factor the equation to solve for x.

(x - 7)(x + 7) = 0

Step 4: Solve for x

Finally, we can solve for x by setting each factor equal to zero.

x - 7 = 0 \Rightarrow x = 7
x + 7 = 0 \Rightarrow x = -7

In this article, we have explored two possible solutions to each of the given equations. We have used algebraic methods to solve these equations and provided step-by-step solutions. The first equation was a quadratic equation, and we used the quadratic formula to solve it. The second equation was also a quadratic equation, and we factored it to solve for x. We have shown that the two possible solutions to each equation are x = 9 and x = 1 for the first equation, and x = 7 and x = -7 for the second equation.
What are the Two Possible Solutions to Each of the Equations Below? - Q&A

In our previous article, we explored two possible solutions to each of the given equations. We used algebraic methods to solve these equations and provided step-by-step solutions. In this article, we will answer some frequently asked questions (FAQs) related to the solutions of the equations.

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

A: A linear equation is an equation in which the highest power of the variable (x) is 1. For example, 2x + 3 = 0 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (x) is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and completing the square. The quadratic formula is a popular method to solve quadratic equations, and it is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by:

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

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, you can plug these values into the quadratic formula to get the solutions.

Q: What are the two possible solutions to the equation 3(x−5)2=483(x-5)^2 = 48?

A: The two possible solutions to the equation 3(x−5)2=483(x-5)^2 = 48 are x = 9 and x = 1.

Q: What are the two possible solutions to the equation 2x2−56=422x^2 - 56 = 42?

A: The two possible solutions to the equation 2x2−56=422x^2 - 56 = 42 are x = 7 and x = -7.

Q: How do I check if my solutions are correct?

A: To check if your solutions are correct, you can plug them back into the original equation to see if they satisfy the equation.

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

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

  • Not identifying the values of a, b, and c correctly
  • Not using the correct formula to solve the equation
  • Not checking if the solutions satisfy the original equation

In this article, we have answered some frequently asked questions (FAQs) related to the solutions of the equations. We have provided step-by-step solutions to the equations and explained the quadratic formula. We have also discussed some common mistakes to avoid when solving quadratic equations. We hope that this article has been helpful in clarifying any doubts you may have had about solving quadratic equations.