Suppose The Quadratic Equation $x^2 + Bx + C = 0$ Has The Solutions $-7 + 1i$ And \$-7 - 1i$[/tex\]. What Are $b$ And $c$?$b = \square$c = \square$

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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will explore how to find the coefficients $b$ and $c$ of a quadratic equation when its solutions are given as complex numbers.

Complex Solutions of a Quadratic Equation

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but with some additional rules.

In the given quadratic equation $x^2 + bx + c = 0$, we are told that the solutions are $-7 + 1i$ and $-7 - 1i$. These are complex numbers, and we can use them to find the values of $b$ and $c$.

Using Vieta's Formulas

Vieta's formulas are a set of equations that relate the coefficients of a polynomial to its roots. For a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$, Vieta's formulas state that:

$r_1 + r_2 = -\frac{b}{a}$

$r_1 \cdot r_2 = \frac{c}{a}$

In our case, the quadratic equation is $x^2 + bx + c = 0$, so $a = 1$. The roots are $-7 + 1i$ and $-7 - 1i$, so we can use Vieta's formulas to find the values of $b$ and $c$.

Finding the Value of b

Using the first Vieta's formula, we have:

$(-7 + 1i) + (-7 - 1i) = -\frac{b}{1}$

Simplifying the left-hand side, we get:

$-14 = -b$

Multiplying both sides by $-1$, we get:

$b = 14$

Finding the Value of c

Using the second Vieta's formula, we have:

$(-7 + 1i) \cdot (-7 - 1i) = \frac{c}{1}$

Expanding the left-hand side, we get:

$49 - i^2 = c$

Since $i^2 = -1$, we have:

$49 + 1 = c$

Simplifying, we get:

$c = 50$

Conclusion

In this article, we have shown how to find the coefficients $b$ and $c$ of a quadratic equation when its solutions are given as complex numbers. We used Vieta's formulas to relate the coefficients to the roots of the equation, and then solved for $b$ and $c$ using the given solutions. The values of $b$ and $c$ are $14$ and $50$, respectively.

Example Use Case

Suppose we want to find the solutions of the quadratic equation $x^2 + 14x + 50 = 0$. We can use the quadratic formula to find the solutions:

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

Substituting $a = 1$, $b = 14$, and $c = 50$, we get:

$x = \frac{-14 \pm \sqrt{14^2 - 4 \cdot 1 \cdot 50}}{2 \cdot 1}$

Simplifying, we get:

$x = \frac{-14 \pm \sqrt{196 - 200}}{2}$

$x = \frac{-14 \pm \sqrt{-4}}{2}$

$x = \frac{-14 \pm 2i}{2}$

$x = -7 \pm i$

$$

Introduction

In our previous article, we explored how to find the coefficients $b$ and $c$ of a quadratic equation when its solutions are given as complex numbers. We used Vieta's formulas to relate the coefficients to the roots of the equation, and then solved for $b$ and $c$ using the given solutions. In this article, we will answer some frequently asked questions about quadratic equations with complex solutions.

Q: What is the difference between a quadratic equation with real solutions and a quadratic equation with complex solutions?

A: A quadratic equation with real solutions has roots that are real numbers, whereas a quadratic equation with complex solutions has roots that are complex numbers. Complex numbers have both real and imaginary parts, and they can be added, subtracted, multiplied, and divided just like real numbers.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can a quadratic equation have both real and complex solutions?

A: No, a quadratic equation cannot have both real and complex solutions. The solutions of a quadratic equation are either real or complex, but not both.

Q: How do I find the solutions of a quadratic equation with complex solutions?

A: To find the solutions of a quadratic equation with complex solutions, you can use the quadratic formula:

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

Substitute the values of $a$, $b$, and $c$ into the formula, and simplify to find the solutions.

Q: What is the difference between the conjugate of a complex number and the complex conjugate of a quadratic equation?

A: The conjugate of a complex number is another complex number with the same real part and the opposite imaginary part. For example, the conjugate of $3 + 4i$ is $3 - 4i$. The complex conjugate of a quadratic equation is a new quadratic equation that has the same coefficients as the original equation, but with the opposite sign of the imaginary part. For example, the complex conjugate of $x^2 + 2x + 3i$ is $x^2 + 2x - 3i$.

Q: Can I use the quadratic formula to find the solutions of a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to find the solutions of a quadratic equation with complex solutions. However, you will need to simplify the expression under the square root to find the solutions.

Q: What is the significance of the imaginary unit $i$ in a quadratic equation with complex solutions?

A: The imaginary unit $i$ is a mathematical constant that satisfies the equation $i^2 = -1$. In a quadratic equation with complex solutions, the imaginary unit $i$ is used to represent the imaginary part of the solutions.

Q: Can I use a calculator to find the solutions of a quadratic equation with complex solutions?

A: Yes, you can use a calculator to find the solutions of a quadratic equation with complex solutions. However, you will need to enter the equation in the correct format and use the calculator's complex number capabilities to find the solutions.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations with complex solutions. We have discussed the difference between quadratic equations with real and complex solutions, how to determine if a quadratic equation has real or complex solutions, and how to find the solutions of a quadratic equation with complex solutions. We have also discussed the significance of the imaginary unit $i$ in a quadratic equation with complex solutions and how to use a calculator to find the solutions of a quadratic equation with complex solutions.