Given The Equation, What Are The Solutions? − 2 ( X − 3 ) 2 = − 20 -2(x-3)^2 = -20 − 2 ( X − 3 ) 2 = − 20 A. X = − 3 + 10 , X = − 3 − 10 X = -3 + \sqrt{10}, X = -3 - \sqrt{10} X = − 3 + 10 ​ , X = − 3 − 10 ​ B. X = 3 + 10 , X = 3 − 10 X = 3 + \sqrt{10}, X = 3 - \sqrt{10} X = 3 + 10 ​ , X = 3 − 10 ​ C. No SolutionD. X = 13 , X = 7 X = 13, X = 7 X = 13 , X = 7

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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 delve into the world of quadratic equations and explore the steps involved in solving them. We will use the given equation, −2(x−3)2=−20-2(x-3)^2 = -20, as a case study to demonstrate the solution process.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is:

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

where a, b, and c are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Given Equation

The given equation is:

−2(x−3)2=−20-2(x-3)^2 = -20

To solve this equation, we need to isolate the variable x. The first step is to expand the squared term:

−2(x2−6x+9)=−20-2(x^2 - 6x + 9) = -20

Expanding and Simplifying

Expanding the squared term, we get:

−2x2+12x−18=−20-2x^2 + 12x - 18 = -20

Next, we add 20 to both sides of the equation to get:

−2x2+12x−2=0-2x^2 + 12x - 2 = 0

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

In our case, a = -2, b = 12, and c = -2. Plugging these values into the quadratic formula, we get:

x=−12±122−4(−2)(−2)2(−2)x = \frac{-12 \pm \sqrt{12^2 - 4(-2)(-2)}}{2(-2)}

Simplifying the Quadratic Formula

Simplifying the expression under the square root, we get:

x=−12±144−16−4x = \frac{-12 \pm \sqrt{144 - 16}}{-4}

x=−12±128−4x = \frac{-12 \pm \sqrt{128}}{-4}

Simplifying the Square Root

The square root of 128 can be simplified as:

128=64×2=82\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}

Substituting the Simplified Square Root

Substituting the simplified square root back into the quadratic formula, we get:

x=−12±82−4x = \frac{-12 \pm 8\sqrt{2}}{-4}

Simplifying the Expression

Simplifying the expression, we get:

x=3±22x = 3 \pm 2\sqrt{2}

Conclusion

The solutions to the given equation are:

x=3+22,x=3−22x = 3 + 2\sqrt{2}, x = 3 - 2\sqrt{2}

These solutions match option B, x=3+10,x=3−10x = 3 + \sqrt{10}, x = 3 - \sqrt{10}.

Discussion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we used the given equation, −2(x−3)2=−20-2(x-3)^2 = -20, as a case study to demonstrate the solution process. We expanded and simplified the equation, used the quadratic formula, and simplified the square root to arrive at the solutions.

Final Answer

The final answer is:

  • x=3+10,x=3−10x = 3 + \sqrt{10}, x = 3 - \sqrt{10}

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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 our previous article, we explored the steps involved in solving quadratic equations using the given equation, −2(x−3)2=−20-2(x-3)^2 = -20. In this article, we will answer some frequently asked questions about quadratic equations to help you better understand this concept.

Q&A

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, x) is two. The general form of a quadratic equation is:

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

where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, the quadratic formula, and graphing. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

x=−b±b2−4ac2ax = \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 states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

Q: What is the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, given by:

b2−4acb^2 - 4ac

If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant. If the discriminant is:

  • Positive, the equation has two distinct solutions.
  • Zero, the equation has one repeated solution.
  • Negative, the equation has no real solutions.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is:

ax+b=0ax + b = 0

where a and b are constants.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions if the discriminant is negative.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we answered some frequently asked questions about quadratic equations to help you better understand this concept. We hope this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Final Answer

The final answer is:

  • Quadratic equations are polynomial equations of degree two.
  • The quadratic formula is a powerful tool for solving quadratic equations.
  • The discriminant determines the number of solutions to a quadratic equation.
  • Quadratic equations can have at most two solutions.
  • Quadratic equations can have no solutions if the discriminant is negative.