To Find Range Of Roots Of Quadratic Equation

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Introduction

In algebra, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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, and xx is the variable. The roots of a quadratic equation are the values of xx that satisfy the equation. In this article, we will discuss how to find the range of roots of a given quadratic equation.

The Quadratic Equation

The given quadratic equation is (k+1)x2−(20k+14)x+91k+40=0(k+1)x^2 - (20k+14)x + 91k +40 =0, where k>0k>0. To find the range of roots, we need to first find the roots of the equation. The roots of a quadratic equation can be found using the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

The Quadratic Formula

The quadratic formula is a fundamental concept in algebra that allows us to find the roots of a quadratic equation. The formula is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. In our case, a=k+1a = k+1, b=−(20k+14)b = -(20k+14), and c=91k+40c = 91k + 40. Plugging these values into the quadratic formula, we get:

x=(20k+14)±(20k+14)2−4(k+1)(91k+40)2(k+1)x = \frac{(20k+14) \pm \sqrt{(20k+14)^2 - 4(k+1)(91k + 40)}}{2(k+1)}

Simplifying the Quadratic Formula

To simplify the quadratic formula, we need to expand and simplify the expression under the square root. Expanding the expression, we get:

(20k+14)2−4(k+1)(91k+40)(20k+14)^2 - 4(k+1)(91k + 40)

=400k2+560k+196−364k−160= 400k^2 + 560k + 196 - 364k - 160

=400k2−104k+36= 400k^2 - 104k + 36

Simplifying the Quadratic Formula (continued)

Now that we have simplified the expression under the square root, we can rewrite the quadratic formula as:

x=(20k+14)±400k2−104k+362(k+1)x = \frac{(20k+14) \pm \sqrt{400k^2 - 104k + 36}}{2(k+1)}

The Discriminant

The expression under the square root in the quadratic formula is called the discriminant. The discriminant is given by b2−4acb^2 - 4ac. In our case, the discriminant is 400k2−104k+36400k^2 - 104k + 36. The discriminant determines the nature of the roots of the quadratic equation. 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.

The Range of Roots

To find the range of roots, we need to consider the possible values of the discriminant. Since the discriminant is 400k2−104k+36400k^2 - 104k + 36, we can see that it is always positive for k>0k>0. Therefore, the equation has two distinct real roots for all values k>0k>0.

Finding the Range of Roots

To find the range of roots, we need to find the minimum and maximum values of the roots. The minimum value of the roots occurs when the discriminant is minimum, and the maximum value of the roots occurs when the discriminant is maximum.

Finding the Minimum Value of the Roots

To find the minimum value of the roots, we need to find the minimum value of the discriminant. The discriminant is given by 400k2−104k+36400k^2 - 104k + 36. To find the minimum value of the discriminant, we can complete the square:

400k2−104k+36400k^2 - 104k + 36

=400(k2−104400k)+36= 400(k^2 - \frac{104}{400}k) + 36

=400(k2−1350k)+36= 400(k^2 - \frac{13}{50}k) + 36

=400(k2−1350k+1691600)−1694+36= 400(k^2 - \frac{13}{50}k + \frac{169}{1600}) - \frac{169}{4} + 36

=400(k−13100)2−1694+36= 400(k - \frac{13}{100})^2 - \frac{169}{4} + 36

Finding the Minimum Value of the Roots (continued)

The minimum value of the discriminant occurs when the term 400(k−13100)2400(k - \frac{13}{100})^2 is zero. Therefore, the minimum value of the discriminant is −1694+36=14-\frac{169}{4} + 36 = \frac{1}{4}. Since the discriminant is always positive for k>0k>0, the minimum value of the discriminant is 14\frac{1}{4}.

Finding the Maximum Value of the Roots

To find the maximum value of the roots, we need to find the maximum value of the discriminant. The discriminant is given by 400k2−104k+36400k^2 - 104k + 36. To find the maximum value of the discriminant, we can complete the square:

400k2−104k+36400k^2 - 104k + 36

=400(k2−104400k)+36= 400(k^2 - \frac{104}{400}k) + 36

=400(k2−1350k)+36= 400(k^2 - \frac{13}{50}k) + 36

=400(k2−1350k+1691600)−1694+36= 400(k^2 - \frac{13}{50}k + \frac{169}{1600}) - \frac{169}{4} + 36

=400(k−13100)2−1694+36= 400(k - \frac{13}{100})^2 - \frac{169}{4} + 36

Finding the Maximum Value of the Roots (continued)

The maximum value of the discriminant occurs when the term 400(k−13100)2400(k - \frac{13}{100})^2 is maximum. Since the term 400(k−13100)2400(k - \frac{13}{100})^2 is always non-negative, the maximum value of the discriminant is unbounded. However, since the discriminant is always positive for k>0k>0, the maximum value of the discriminant is unbounded.

Conclusion

In conclusion, the range of roots of the given quadratic equation is all real numbers for k>0k>0. The minimum value of the roots is 14\frac{1}{4}, and the maximum value of the roots is unbounded.

References

  • [1] "Quadratic Equation" by Wikipedia
  • [2] "Quadratic Formula" by Math Open Reference
  • [3 "Discriminant" by Math Is Fun

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Introduction

In our previous article, we discussed how to find the range of roots of a given quadratic equation. In this article, we will answer some frequently asked questions related to the topic.

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 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, and xx is the variable.

Q: How do I find the roots of a quadratic equation?

A: To find the roots of a quadratic equation, you can use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. This formula will give you two roots, which are the values of xx that satisfy the equation.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by b2−4acb^2 - 4ac. The discriminant determines the nature of the roots of the quadratic equation. 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 find the range of roots of a quadratic equation?

A: To find the range of roots of a quadratic equation, you need to find the minimum and maximum values of the roots. The minimum value of the roots occurs when the discriminant is minimum, and the maximum value of the roots occurs when the discriminant is maximum.

Q: What is the minimum value of the roots?

A: The minimum value of the roots is the smallest value that the roots can take. This value occurs when the discriminant is minimum. In the case of the quadratic equation (k+1)x2−(20k+14)x+91k+40=0(k+1)x^2 - (20k+14)x + 91k +40 =0, the minimum value of the roots is 14\frac{1}{4}.

Q: What is the maximum value of the roots?

A: The maximum value of the roots is the largest value that the roots can take. This value occurs when the discriminant is maximum. In the case of the quadratic equation (k+1)x2−(20k+14)x+91k+40=0(k+1)x^2 - (20k+14)x + 91k +40 =0, the maximum value of the roots is unbounded.

Q: Can I find the range of roots of a quadratic equation with complex roots?

A: Yes, you can find the range of roots of a quadratic equation with complex roots. However, the range of roots will be complex numbers, not real numbers.

Q: How do I apply the quadratic formula to a quadratic equation with complex roots?

A: To apply the quadratic formula to a quadratic equation with complex roots, you need to use the complex conjugate of the square root of the discriminant. The complex conjugate of a complex number a+bia+bi is a−bia-bi.

Q: What is the significance of range of roots of a quadratic equation?

A: The range of roots of a quadratic equation is significant because it gives you information about the behavior of the equation. For example, if the range of roots is all real numbers, the equation has two distinct real roots. If the range of roots is complex numbers, the equation has complex roots.

Conclusion

In conclusion, the range of roots of a quadratic equation is an important concept in algebra. It gives you information about the behavior of the equation and helps you to understand the nature of the roots. We hope that this Q&A article has been helpful in answering your questions about the range of roots of a quadratic equation.

References

  • [1] "Quadratic Equation" by Wikipedia
  • [2] "Quadratic Formula" by Math Open Reference
  • [3 "Discriminant" by Math Is Fun