If One Zero Of A Quadratic Polynomial { K X^2+3 X+k } Is 2, Then The Value Of { K } Is:A. { \frac{5}{6} } B. { − 5 6 -\frac{5}{6} − 6 5 ​ $}$C. { \frac{6}{5} } D. { -\frac{6}{5} }

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Introduction

In algebra, a quadratic polynomial is a polynomial of degree two, which means the highest power of the variable is two. The general form of a quadratic polynomial is ${ax^2 + bx + c}$, where ${a}$, ${b}$, and ${c}$ are constants. In this article, we will focus on a specific quadratic polynomial, ${kx^2 + 3x + k}$, and find the value of ${k}$ given that one of its zeros is 2.

Understanding Zeros of a Quadratic Polynomial

The zeros of a quadratic polynomial are the values of ${x}$ that make the polynomial equal to zero. In other words, if we substitute a zero of the polynomial into the equation, the result will be zero. For a quadratic polynomial of the form ${ax^2 + bx + c}$, the zeros can be found using the quadratic formula:

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

However, in this case, we are given that one of the zeros is 2, so we can use this information to find the value of ${k}$.

Using the Given Zero to Find the Value of ${k}$

Since one of the zeros is 2, we can substitute ${x = 2}$ into the equation ${kx^2 + 3x + k = 0}$ to get:

${k(2)^2 + 3(2) + k = 0}$

Simplifying the equation, we get:

${4k + 6 + k = 0}$

Combine like terms:

${5k + 6 = 0}$

Subtract 6 from both sides:

${5k = -6}$

Divide both sides by 5:

${k = -\frac{6}{5}}$

Therefore, the value of ${k}$ is ${-\frac{6}{5}}$.

Conclusion

In this article, we used the given zero of a quadratic polynomial to find the value of ${k}$. We started with the general form of a quadratic polynomial and substituted the given zero into the equation to get a linear equation in terms of ${k}$. We then solved for ${k}$ to find the value of ${-\frac{6}{5}}$.

Answer

The value of ${k}$ is ${-\frac{6}{5}}$.

Comparison with Options

Comparing our answer with the given options, we can see that the correct answer is:

B. ${-\frac{5}{6}}$

However, we found that the value of ${k}$ is ${-\frac{6}{5}}$, which is not among the options. This suggests that there may be a mistake in the options or in our calculation.

Revisiting the Calculation

Let's revisit the calculation to see if we made any mistakes:

${k(2)^2 + 3(2) + k = 0}$

Simplifying the equation, we get:

${4k + 6 + k = 0}$

Combine like terms:

${5k + 6 = 0}$

Subtract 6 from both sides:

${5k = -6}$

Divide both sides by 5:

${k = -\frac{6}{5}}$

However, we can see that the correct answer is actually ${-\frac{6}{5}}$, but it is not among the options. This suggests that the options may be incorrect.

Conclusion

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Q: What is a quadratic polynomial?

A: A quadratic polynomial is a polynomial of degree two, which means the highest power of the variable is two. The general form of a quadratic polynomial is ${ax^2 + bx + c}$, where ${a}$, ${b}$, and ${c}$ are constants.

Q: How do you find the zeros of a quadratic polynomial?

A: The zeros of a quadratic polynomial can be found using the quadratic formula:

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

However, if one of the zeros is given, we can use that information to find the value of the other constants.

Q: How do you find the value of ${k}$ in a quadratic polynomial given one of its zeros?

A: To find the value of ${k}$ in a quadratic polynomial given one of its zeros, we can substitute the given zero into the equation and solve for ${k}$. For example, if one of the zeros is 2, we can substitute ${x = 2}$ into the equation ${kx^2 + 3x + k = 0}$ to get:

${k(2)^2 + 3(2) + k = 0}$

Simplifying the equation, we get:

${4k + 6 + k = 0}$

Combine like terms:

${5k + 6 = 0}$

Subtract 6 from both sides:

${5k = -6}$

Divide both sides by 5:

${k = -\frac{6}{5}}$

Q: What if the options for the value of ${k}$ are not among the possible answers?

A: If the options for the value of ${k}$ are not among the possible answers, it may be a mistake in the options or in our calculation. We should revisit the calculation to see if we made any mistakes.

Q: How do you know if the options are correct or not?

A: To know if the options are correct or not, we should check our calculation carefully and make sure that we did not make any mistakes. We should also check if the options are consistent with the given information.

Q: What if the quadratic polynomial has complex zeros?

A: If the quadratic polynomial has complex zeros, we can use the quadratic formula to find the zeros. The quadratic formula is:

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

However, if the discriminant ${b^2 - 4ac}$ is negative, the zeros will be complex.

Q: How do you find the value of ${k}$ in a quadratic polynomial with complex zeros?

A: To find the value of ${k}$ in a quadratic polynomial with complex zeros, we can use the quadratic formula to find the zeros and then substitute the zeros into the equation to get a linear equation in terms of ${k}$. We can then solve for ${k}$ to find the value.

Q: What if the quadratic polynomial has no real zeros?

A: If the quadratic polynomial has no real zeros, it means that the discriminant ${b^2 - 4ac}$ is negative. In this case, the zeros will be complex, and we can use the quadratic formula to find the zeros.

Conclusion

In this article, we answered some common questions about quadratic polynomials, including how to find the zeros, how to find the value of ${k}$ given one of its zeros, and how to handle complex zeros. We also discussed how to check if the options are correct or not and how to find the value of ${k}$ in a quadratic polynomial with complex zeros or no real zeros.