Solve The Equation:$3x^2 + 1 - 7x = 13x^2$Choose One Answer:A. $x = \frac{2 \pm \sqrt{29}}{5}$B. $x = \frac{7 \pm \sqrt{89}}{-20}$C. $x = \frac{-3 \pm \sqrt{105}}{-12}$D. $x = \frac{-1 \pm \sqrt{73}}{9}$

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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 specific quadratic equation, 3x2+1−7x=13x23x^2 + 1 - 7x = 13x^2, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

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. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

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}

This formula is derived from the process of completing the square, which involves manipulating the equation to create a perfect square trinomial.

Solving the Given Equation

Now, let's apply the quadratic formula to the given equation, 3x2+1−7x=13x23x^2 + 1 - 7x = 13x^2. First, we need to rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. We can do this by subtracting 13x213x^2 from both sides of the equation:

3x2−13x2+1−7x=03x^2 - 13x^2 + 1 - 7x = 0

Simplifying the equation, we get:

−10x2−7x+1=0-10x^2 - 7x + 1 = 0

Now, we can identify the values of aa, bb, and cc:

a=−10a = -10 b=−7b = -7 c=1c = 1

Applying the Quadratic Formula

Substituting the values of aa, bb, and cc into the quadratic formula, we get:

x=−(−7)±(−7)2−4(−10)(1)2(−10)x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(-10)(1)}}{2(-10)}

Simplifying the expression, we get:

x=7±49+40−20x = \frac{7 \pm \sqrt{49 + 40}}{-20}

x=7±89−20x = \frac{7 \pm \sqrt{89}}{-20}

Conclusion

In this article, we solved the quadratic equation 3x2+1−7x=13x23x^2 + 1 - 7x = 13x^2 using the quadratic formula. We identified the values of aa, bb, and cc and substituted them into the formula to find the solutions. The solutions are given by:

x=7±89−20x = \frac{7 \pm \sqrt{89}}{-20}

This is the correct answer, which corresponds to option B.

Final Answer

The final answer is:

  • B. x=7±89−20x = \frac{7 \pm \sqrt{89}}{-20}

This answer is the correct solution to the given quadratic equation.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, including their definition, properties, and methods for 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: What are the Properties of a Quadratic Equation?

A: Quadratic equations have several important properties, including:

  • The sum of the roots: The sum of the roots of a quadratic equation is equal to −b/a-b/a.
  • The product of the roots: The product of the roots of a quadratic equation is equal to c/ac/a.
  • The discriminant: The discriminant of a quadratic equation is given by b2−4acb^2 - 4ac. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods for solving quadratic equations, including:

  • Factoring: If the quadratic expression can be factored into the product of two binomials, we can solve the equation by setting each factor equal to zero.
  • Completing the square: This method involves manipulating the equation to create a perfect square trinomial.
  • The quadratic formula: This 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}

Q: What is the Quadratic Formula?

A: 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}

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to identify the values of aa, bb, and cc in the equation. Then, you can substitute these values into the formula to find the solutions.

Q: What is the Discriminant?

A: The discriminant of a quadratic equation is given by b2−4acb^2 - 4ac. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: How Do I Determine the Number of Real Roots?

A: To determine the number of real roots, you need to examine the discriminant. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: What is the Relationship Between the Roots and the Coefficients?

A: The roots of a quadratic equation are related to the coefficients of the equation. The sum of the roots is equal to −b/a-b/a, and the product of the roots is equal to c/ac/a.

Conclusion

In this article, we addressed some of the most frequently asked questions about quadratic equations, including their definition, properties, and methods for solving them. We also discussed the quadratic formula, the discriminant, and the relationship between the roots and the coefficients. By understanding these concepts, you can solve quadratic equations with confidence.

Final Answer

The final answer is:

  • B. x=7±89−20x = \frac{7 \pm \sqrt{89}}{-20}

This answer is the correct solution to the given quadratic equation.