Select The Correct Answer.Find The Solutions For $x$ In The Equation Below.$x^2 - 9x + 20 = 0$A. $ X = 4 ; X = − 5 X = 4; X = -5 X = 4 ; X = − 5 [/tex] B. $x = 4; X = 5$ C. $x = -4; X = -5$ D. $x = -4; X =

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Introduction

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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 quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the given equation $x^2 - 9x + 20 = 0$ as an example to demonstrate the steps involved in solving quadratic equations.

Understanding Quadratic Equations

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A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$. The solutions to a quadratic equation are called roots or zeros.

The Quadratic Formula

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The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula is derived from the fact that a quadratic equation can be factored as:

$$(x - r_1)(x - r_2) = 0$$

where $r_1$ and $r_2$ are the roots of the equation.

Solving the Given Equation

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Now, let's apply the quadratic formula to the given equation $x^2 - 9x + 20 = 0$. We have:

$$a = 1, b = -9, c = 20$$

Substituting these values into the quadratic formula, we get:

$$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(20)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{9 \pm \sqrt{81 - 80}}{2}$$

$$x = \frac{9 \pm \sqrt{1}}{2}$$

$$x = \frac{9 \pm 1}{2}$$

Therefore, we have two possible solutions:

$$x = \frac{9 + 1}{2} = 5$$

$$x = \frac{9 - 1}{2} = 4$$

Conclusion

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In this article, we have demonstrated the steps involved in solving quadratic equations using the quadratic formula. We have applied the formula to the given equation $x^2 - 9x + 20 = 0$ and obtained the solutions $x = 5$ and $x = 4$. These solutions are the correct answers to the given equation.

Final Answer

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The final answer is:

• A. $x = 4; x = -5$ is incorrect.
• B. $x = 4; x = 5$ is correct.
• C. $x = -4; x = -5$ is incorrect.
• D. $x = -4; x = 5$ is incorrect.

The correct answer is B. $x = 4; x = 5$.

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Introduction

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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 address some of the most frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

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A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$. The solutions to a quadratic equation are called roots or zeros.

Q: What is the quadratic formula?

---

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula is derived from the fact that a quadratic equation can be factored as:

$$(x - r_1)(x - r_2) = 0$$

where $r_1$ and $r_2$ are the roots of the equation.

Q: How do I apply the quadratic formula?

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To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, substitute these values into the quadratic formula and simplify the expression.

Q: What are the steps involved in solving a quadratic equation?

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The steps involved in solving a quadratic equation are:

1. Write the equation in standard form: Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Identify the values of $a$, $b$, and $c$: Identify the values of $a$, $b$, and $c$ in the quadratic equation.
3. Apply the quadratic formula: Substitute the values of $a$, $b$, and $c$ into the quadratic formula and simplify the expression.
4. Solve for $x$: Solve for $x$ by simplifying the expression.

Q: What are the different types of solutions to a quadratic equation?

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The different types of solutions to a quadratic equation are:

• Real and distinct solutions: These are solutions that are real numbers and are distinct from each other.
• Real and repeated solutions: These are solutions that are real numbers and are repeated.
• Complex solutions: These are solutions that are complex numbers.

Q: How do I determine the number of solutions to a quadratic equation?

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To determine the number of solutions to a quadratic equation, you need to examine the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is:

• Positive: The equation has two distinct real solutions.
• Zero: The equation has one real solution.
• Negative: The equation has no real solutions.

Q: What are some common mistakes to avoid when solving quadratic equations?

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Some common mistakes to avoid when solving quadratic equations are:

• Not writing the equation in standard form: Make sure to write the equation in the form $ax^2 + bx + c = 0$.
• Not identifying the values of $a$, $b$, and $c$: Make sure to identify the values of $a$, $b$, and $c$ in the quadratic equation.
• Not applying the quadratic formula correctly: Make sure to substitute the values of $a$, $b$, and $c$ into the quadratic formula and simplify the expression correctly.

Conclusion

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In this article, we have addressed some of the most frequently asked questions about quadratic equations. We have covered topics such as the definition of a quadratic equation, the quadratic formula, and the steps involved in solving a quadratic equation. We have also discussed the different types of solutions to a quadratic equation and how to determine the number of solutions. Finally, we have highlighted some common mistakes to avoid when solving quadratic equations.