Solve For { X $}$.1) ${ X^2 + 3x + 2 }$2) ${ X^2 - 7x + 12 }$3) ${ 3x^2 + 2x - 8 }$4) ${ 5w^2 - 9w - 2 }$5) ${ 2x(3x - 3) = 0 }$6) ${ (x - 3)(6x - 2) = 0 }$

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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 delve into the world of quadratic equations and provide a step-by-step guide on how to solve them. We will cover various types of quadratic equations, including those with real and complex roots, and provide examples to illustrate each concept.

What are Quadratic Equations?

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

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

where $a$, $b$, and $c$ are constants, and $a$ cannot be zero. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and completing the square.

Method 1: Factoring

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Factoring is a simple and effective method for solving quadratic equations. It involves expressing the quadratic equation as a product of two binomials. The general form of a factored quadratic equation is:

$$(x + p)(x + q) = 0$$

where $p$ and $q$ are constants. To factor a quadratic equation, we need to find two numbers whose product is $c$ (the constant term) and whose sum is $b$ (the coefficient of the linear term).

Example 1: $x^2 + 3x + 2 = 0$

To factor this equation, we need to find two numbers whose product is $2$ and whose sum is $3$. The numbers are $1$ and $2$, so we can write the equation as:

$$(x + 1)(x + 2) = 0$$

This equation has two solutions: $x = -1$ and $x = -2$.

Example 2: $x^2 - 7x + 12 = 0$

To factor this equation, we need to find two numbers whose product is $12$ and whose sum is $-7$. The numbers are $-3$ and $-4$, so we can write the equation as:

$$(x - 3)(x - 4) = 0$$

This equation has two solutions: $x = 3$ and $x = 4$.

Method 2: The Quadratic Formula

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The quadratic formula is a powerful tool for solving quadratic equations. It is a formula that gives the solutions to a quadratic equation in the form of:

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

where $a$, $b$, and $c$ are the constants in the quadratic equation.

Example 3: $3x^2 + 2x - 8 = 0$

To solve this equation using the quadratic formula, we need to plug in the values of $a$, $b$, and $c$ into the formula:

$$x = \frac{-2 \pm \sqrt{2^2 - 4(3)(-8)}}{2(3)}$$

Simplifying the expression, we get:

$$x = \frac{-2 \pm \sqrt{100}}{6}$$

$$x = \frac{-2 \pm 10}{6}$$

This equation has two solutions: $x = 1$ and $x = -3$.

Method 3: Completing the Square

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Completing the square is a method for solving quadratic equations by rewriting them in a perfect square form. The general form of a quadratic equation that can be completed to a square is:

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

where $b$ and $c$ are constants.

Example 4: $5w^2 - 9w - 2 = 0$

To complete the square for this equation, we need to add and subtract $(\frac{b}{2})^2$ to the equation:

$$5w^2 - 9w - 2 = 0$$

$$5w^2 - 9w + (\frac{-9}{2})^2 - (\frac{-9}{2})^2 - 2 = 0$$

$$5w^2 - 9w + \frac{81}{4} - \frac{81}{4} - 2 = 0$$

$$5(w^2 - \frac{9}{5}w + \frac{81}{20}) - \frac{81}{4} - 2 = 0$$

$$5(w - \frac{9}{10})^2 - \frac{81}{4} - 2 = 0$$

$$5(w - \frac{9}{10})^2 = \frac{81}{4} + 2$$

$$5(w - \frac{9}{10})^2 = \frac{85}{4}$$

$$w - \frac{9}{10} = \pm \sqrt{\frac{85}{20}}$$

$$w = \frac{9}{10} \pm \sqrt{\frac{85}{20}}$$

This equation has two solutions: $w = \frac{9}{10} + \sqrt{\frac{85}{20}}$ and $w = \frac{9}{10} - \sqrt{\frac{85}{20}}$.

Solving Equations with Complex Roots

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Quadratic equations can have complex roots, which are roots that involve imaginary numbers. Complex roots occur when the discriminant ($b^2 - 4ac$) is negative.

Example 5: $2x(3x - 3) = 0$

To solve this equation, we need to set each factor equal to zero:

$$2x = 0$$

$$3x - 3 = 0$$

Solving for $x$, we get:

$$x = 0$$

$$x = 1$$

This equation has two solutions: $x = 0$ and $x = 1$.

Example 6: $(x - 3)(6x - 2) = 0$

To solve this equation, we need to set each factor equal to zero:

$$x - 3 = 0$$

$$6x - 2 = 0$$

Solving for $x$, we get:

$$x = 3$$

$$x = \frac{1}{3}$$

This equation has two solutions: $x = 3$ and $x = \frac{1}{3}$.

Conclusion

Solving quadratic equations is a crucial skill for students and professionals alike. In this article, we have covered various methods for solving quadratic equations, including factoring, the quadratic formula, and completing the square. We have also discussed how to solve equations with complex roots. By mastering these methods, you will be able to solve a wide range of quadratic equations and become proficient in algebra.

Final Tips

• Always check your solutions by plugging them back into the original equation.
• Use the quadratic formula when the equation cannot be factored easily.
• Completing the square is a useful method for solving quadratic equations, but it can be time-consuming.
• Complex roots occur when the discriminant is negative.
• Always simplify your solutions to the simplest form possible.

By following these tips and practicing regularly, you will become proficient in solving quadratic equations and be able to tackle a wide range of algebraic problems.

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Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations and provide detailed answers to help you better understand this topic.

Q1: 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 (usually $x$) is two. The general form of a quadratic equation is:

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

where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

Q2: How do I solve a quadratic equation?

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There are several methods for solving quadratic equations, including factoring, the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Factoring

Factoring is a simple and effective method for solving quadratic equations. It involves expressing the quadratic equation as a product of two binomials. The general form of a factored quadratic equation is:

$$(x + p)(x + q) = 0$$

where $p$ and $q$ are constants.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is a formula that gives the solutions to a quadratic equation in the form of:

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

where $a$, $b$, and $c$ are the constants in the quadratic equation.

Completing the Square

Completing the square is a method for solving quadratic equations by rewriting them in a perfect square form. The general form of a quadratic equation that can be completed to a square is:

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

where $b$ and $c$ are constants.

Q3: What is the discriminant, and how is it used in quadratic equations?

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The discriminant is a value that is used to determine the nature of the roots of a quadratic equation. It is calculated as:

$$b^2 - 4ac$$

If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.

Q4: How do I determine the nature of the roots of a quadratic equation?

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To determine the nature of the roots of a quadratic equation, you can use the discriminant. If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.

Q5: What is the difference between a quadratic equation and a linear equation?

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A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is:

$$ax + b = 0$$

where $a$ and $b$ are constants.

Q6: Can a quadratic equation have more than two solutions?

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No, a quadratic equation can have at most two solutions. This is because the equation is of degree two, and the highest power of the variable is two.

Q7: How do I check my solutions to a quadratic equation?

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To check your solutions to a quadratic equation, you can plug them back into the original equation. If the solutions satisfy the equation, then they are correct.

Q8: What is the significance of the quadratic formula?

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The quadratic formula is a powerful tool for solving quadratic equations. It is a formula that gives the solutions to a quadratic equation in the form of:

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

where $a$, $b$, and $c$ are the constants in the quadratic equation.

Q9: Can a quadratic equation have complex roots?

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Yes, a quadratic equation can have complex roots. Complex roots occur when the discriminant is negative.

Q10: How do I simplify complex roots?

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To simplify complex roots, you can use the following formula:

$$a + bi = \sqrt{a^2 + b^2}(\cos \theta + i \sin \theta)$$

where $a$ and $b$ are the real and imaginary parts of the complex number, and $\theta$ is the angle of the complex number.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided detailed answers to help you better understand this topic. By mastering the concepts and techniques presented in this article, you will be able to solve a wide range of quadratic equations and become proficient in algebra.

Final Tips

• Always check your solutions by plugging them back into the original equation.
• Use the quadratic formula when the equation cannot be factored easily.
• Completing the square is a useful method for solving quadratic equations, but it can be time-consuming.
• Complex roots occur when the discriminant is negative.
• Always simplify your solutions to the simplest form possible.

By following these tips and practicing regularly, you will become proficient in solving quadratic equations and be able to tackle a wide range of algebraic problems.