If $f(x)=0$ Has Roots $x=\frac{-5 \pm \sqrt{3-12 K^2}}{4}$, For Which Values Of \$k$[/tex\] Will The Roots Be Equal?

Feb 26, 2025 by ADMIN 122 views

$If $f(x)=0$ has roots $x=\frac{-5 \pm \sqrt{3-12 k^2}}{4}$, for which values of \$k$[/tex\] will the roots be equal?$

Introduction

In algebra, the roots of a quadratic equation are the values of the variable that satisfy the equation. When the roots of a quadratic equation are equal, it means that the equation has a repeated root, and the graph of the related function is a parabola that touches the x-axis at a single point. In this article, we will explore the conditions under which the roots of the given quadratic equation will be equal.

The Quadratic Formula

The given quadratic equation is $f(x)=0$, and its roots are given by the quadratic formula:

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

In this case, the coefficients of the quadratic equation are:

a=1

b=-5

c=-12k^2+3

Substituting these values into the quadratic formula, we get:

x=\frac{-(-5) \pm \sqrt{(-5)^2-4(1)(-12k^2+3)}}{2(1)}

Simplifying the expression, we get:

x=\frac{5 \pm \sqrt{25+48k^2-12}}{2}

x=\frac{5 \pm \sqrt{13+48k^2}}{2}

x=\frac{5 \pm \sqrt{48k^2+13}}{2}

x=\frac{5 \pm \sqrt{3-12k^2}}{4}

Equal Roots

For the roots of the quadratic equation to be equal, the discriminant (the expression under the square root) must be equal to zero. In other words, we must have:

48k^2+13=0

Subtracting 13 from both sides, we get:

48k^2=-13

Dividing both sides by 48, we get:

k^2=-\frac{13}{48}

Taking the square root of both sides, we get:

k=\pm\sqrt{-\frac{13}{48}}

However, since the square of any real number is non-negative, we cannot have a real value of k that satisfies this equation. Therefore, the roots of the quadratic equation will never be equal.

Conclusion

In conclusion, the roots of the given quadratic equation will never be equal, regardless of the value of k. This is because the discriminant (the expression under the square root) is always positive, and therefore, the quadratic formula will always yield two distinct roots.

References

[1] "Quadratic Formula" by Math Open Reference. Math Open Reference. Retrieved 2023-02-26.
[2] "Discriminant" by Wolfram MathWorld. Wolfram MathWorld. Retrieved 2023-02-26.

Q&A

Q: What is the condition for the roots of a quadratic equation to be equal?

A: The condition for the roots of a quadratic equation to be equal is that the discriminant (the expression under the square root) must be equal to zero.

Q: How do we find the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is found by subtracting 4ac from b^2, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the discriminant of the given quadratic equation?

A: The discriminant of the given quadratic equation is 3-12k^2.

Q: What is the condition for the roots of the given quadratic equation to be equal?

A: The condition for the roots of the given quadratic equation to be equal is that 3-12k^2 must be equal to zero.

Q: What is the value of k that satisfies the condition for the roots to be equal?

A: Unfortunately, there is no real value of k that satisfies the condition for the roots to be equal, because 3-12k^2 is always positive.

Q: What is the implication of the roots not being equal?

A: The implication of the roots not being equal is that the quadratic equation has two distinct roots, and the graph of the related function is a parabola that does not touch the x-axis at a single point.

Q: What is the relationship between the discriminant and the roots of a quadratic equation?

A: The discriminant is related to the roots of a quadratic equation by the quadratic formula. If the discriminant is positive, the quadratic equation has two distinct roots. If the discriminant is zero, the quadratic equation has one repeated root. If the discriminant is negative, the quadratic equation has no real roots.

Q: What is the significance of the quadratic formula in finding the roots of a quadratic equation?

A: The quadratic formula is a powerful tool for finding the roots of a quadratic equation. It allows us to find the roots of a quadratic equation in terms of the coefficients of the equation.

Q: Can we use the quadratic formula to find the roots of the given quadratic equation?

A: Yes, we can use the quadratic formula to find the roots of the given quadratic equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -5, and c = -12k^2 + 3.

Q: What is the final answer to the problem?

A: Unfortunately, there is no real value of k that satisfies the condition for the roots to be equal. The roots of the quadratic equation will never be equal.