Which Are The Solutions Of The Quadratic Equation?${ X^2 = -5x - 3 }$A. { -5, 0$}$B. { \frac{-5-\sqrt{13}}{2}, \frac{-5+\sqrt{13}}{2}$}$C. { \frac{5-\sqrt{13}}{2}, \frac{5+\sqrt{13}}{2}$}$D. ${ 5, 0\$}

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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 the quadratic equation x2=โˆ’5xโˆ’3x^2 = -5x - 3. We will break down the solution process into manageable steps and provide a clear explanation of each step.

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

In our given equation, x2=โˆ’5xโˆ’3x^2 = -5x - 3, we can rewrite it in the standard form by moving all terms to one side: x2+5x+3=0x^2 + 5x + 3 = 0. Now, we have a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0, where a=1a = 1, b=5b = 5, and c=3c = 3.

The Quadratic Formula

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

x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our case, a=1a = 1, b=5b = 5, and c=3c = 3. Plugging these values into the quadratic formula, we get:

x=โˆ’5ยฑ52โˆ’4(1)(3)2(1)x = \frac{-5 \pm \sqrt{5^2 - 4(1)(3)}}{2(1)}

Simplifying the Quadratic Formula

Now, let's simplify the expression under the square root:

52โˆ’4(1)(3)=25โˆ’12=135^2 - 4(1)(3) = 25 - 12 = 13

So, the quadratic formula becomes:

x=โˆ’5ยฑ132x = \frac{-5 \pm \sqrt{13}}{2}

The Solutions of the Quadratic Equation

Now that we have simplified the quadratic formula, we can find the solutions of the quadratic equation. The solutions are given by:

x=โˆ’5โˆ’132x = \frac{-5 - \sqrt{13}}{2} and x=โˆ’5+132x = \frac{-5 + \sqrt{13}}{2}

These are the two solutions of the quadratic equation x2=โˆ’5xโˆ’3x^2 = -5x - 3.

Conclusion

In this article, we have solved the quadratic equation x2=โˆ’5xโˆ’3x^2 = -5x - 3 using the quadratic formula. We have broken down the solution process into manageable steps and provided a clear explanation of each step. The solutions of the quadratic equation are given by:

x=โˆ’5โˆ’132x = \frac{-5 - \sqrt{13}}{2} and x=โˆ’5+132x = \frac{-5 + \sqrt{13}}{2}

We hope this article has provided a clear understanding of how to solve quadratic equations using the quadratic formula.

Comparison of Solutions

Let's compare the solutions we obtained with the options provided:

A. [โˆ’5,0][-5, 0] B. [โˆ’5โˆ’132,โˆ’5+132][\frac{-5-\sqrt{13}}{2}, \frac{-5+\sqrt{13}}{2}] C. [5โˆ’132,5+132][\frac{5-\sqrt{13}}{2}, \frac{5+\sqrt{13}}{2}] D. [5,0][5, 0]

Our solutions match option B: [โˆ’5โˆ’132,โˆ’5+132][\frac{-5-\sqrt{13}}{2}, \frac{-5+\sqrt{13}}{2}].

Final Answer

The final answer is:

B. [โˆ’5โˆ’132,โˆ’5+132][\frac{-5-\sqrt{13}}{2}, \frac{-5+\sqrt{13}}{2}]

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Introduction

In our previous article, we solved the quadratic equation x2=โˆ’5xโˆ’3x^2 = -5x - 3 using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights into solving them.

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, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

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

x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is given by:

x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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. For example, if you have the equation x2+5x+3=0x^2 + 5x + 3 = 0, you would plug in a=1a = 1, b=5b = 5, and c=3c = 3 into the formula.

Q: What is the difference between the two solutions of a quadratic equation?

A: The two solutions of a quadratic equation are given by the quadratic formula:

x=โˆ’bโˆ’b2โˆ’4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a} and x=โˆ’b+b2โˆ’4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

The difference between the two solutions is the sign of the square root term. If the square root term is positive, the solutions are real and distinct. If the square root term is negative, the solutions are complex and conjugate.

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

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula gives two distinct solutions for any quadratic equation.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the discriminant (b2โˆ’4acb^2 - 4ac) is negative.

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

A: To determine the number of solutions of a quadratic equation, you need to calculate the discriminant (b2โˆ’4acb^2 - 4ac). If the discriminant is:

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

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations and provided additional insights into solving them. We hope this article has provided a clear understanding of quadratic equations and how to solve them using the quadratic formula.

Additional Resources

For more information on quadratic equations, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equation Solver
  • Wolfram Alpha: Quadratic Equation Solver

Final Answer

The final answer is:

  • The quadratic formula is a powerful tool for solving quadratic equations.
  • The quadratic formula is given by: x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • The two solutions of a quadratic equation are given by the quadratic formula.
  • A quadratic equation can have at most two solutions.
  • A quadratic equation can have no solutions if the discriminant is negative.