Solving Quadratic Equations Rewriting To Standard Form And Finding Real Solutions

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In this comprehensive guide, we will delve into the process of rewriting quadratic equations into their standard form and subsequently finding their real solutions. Quadratic equations, which are polynomial equations of the second degree, play a pivotal role in various fields of mathematics, physics, engineering, and economics. Mastering the techniques to manipulate and solve these equations is crucial for anyone seeking a strong foundation in these disciplines. We will systematically address four distinct quadratic equations, meticulously transforming them into their standard form and employing appropriate methods to determine their real roots. This step-by-step approach will not only solidify your understanding of quadratic equations but also equip you with the skills to tackle a wide range of related problems.

Understanding the Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The coefficient 'a' cannot be zero, as this would reduce the equation to a linear form. Rewriting a quadratic equation into this standard form is a crucial first step in solving it, as it allows us to readily identify the coefficients 'a', 'b', and 'c', which are essential for applying various solution methods such as factoring, completing the square, or using the quadratic formula. The standard form provides a consistent structure that simplifies the process of analyzing and solving quadratic equations, making it easier to determine the nature and value of the roots.

Methods for Solving Quadratic Equations

Several methods can be employed to find the real solutions (or roots) of a quadratic equation. The most common methods include:

  • Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If the equation can be factored, the solutions can be found by setting each factor equal to zero and solving for 'x'. Factoring is generally the quickest method when applicable, but it is not always possible to factor a quadratic equation easily.

  • Completing the Square: This technique involves manipulating the equation to create a perfect square trinomial on one side, which can then be factored as a squared binomial. Completing the square is a versatile method that can be used to solve any quadratic equation, and it also forms the basis for deriving the quadratic formula.

  • Quadratic Formula: The quadratic formula is a general solution that can be applied to any quadratic equation in standard form. It provides a direct way to find the solutions, regardless of whether the equation can be factored or not. The quadratic formula is given by:

    x = (-b ± √(b² - 4ac)) / 2a

    The discriminant (b² - 4ac) within the quadratic formula plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). If it is negative, the equation has no real roots, but it does have two complex roots.

Solving the Given Quadratic Equations

Now, let's apply these concepts to rewrite the given quadratic equations into their standard form and find their real solutions.

1. 5x² + 2x = -4

To rewrite this equation in standard form, we need to move the constant term to the left side:

5x² + 2x + 4 = 0

Here, a = 5, b = 2, and c = 4. Let's use the quadratic formula to find the solutions:

x = (-b ± √(b² - 4ac)) / 2a x = (-2 ± √(2² - 4 * 5 * 4)) / (2 * 5) x = (-2 ± √(4 - 80)) / 10 x = (-2 ± √(-76)) / 10

Since the discriminant (b² - 4ac = -76) is negative, this equation has no real solutions. The roots are complex numbers.

2. -3x² + 8x = 10

First, rewrite the equation in standard form:

-3x² + 8x - 10 = 0

Here, a = -3, b = 8, and c = -10. Applying the quadratic formula:

x = (-8 ± √(8² - 4 * -3 * -10)) / (2 * -3) x = (-8 ± √(64 - 120)) / -6 x = (-8 ± √(-56)) / -6

Again, the discriminant (b² - 4ac = -56) is negative, indicating that this equation also has no real solutions. The solutions are complex numbers.

3. 10x² = 5

Rewrite in standard form:

10x² - 5 = 0

Here, a = 10, b = 0, and c = -5. Applying the quadratic formula:

x = (-0 ± √(0² - 4 * 10 * -5)) / (2 * 10) x = (± √(200)) / 20 x = (± 10√2) / 20 x = ± √2 / 2

Therefore, the equation has two real solutions: x = √2 / 2 and x = -√2 / 2.

4. 6x(x + 5) = 15

First, expand and rewrite in standard form:

6x² + 30x = 15 6x² + 30x - 15 = 0

Here, a = 6, b = 30, and c = -15. We can simplify the equation by dividing all terms by 3:

2x² + 10x - 5 = 0

Now, a = 2, b = 10, and c = -5. Applying the quadratic formula:

x = (-10 ± √(10² - 4 * 2 * -5)) / (2 * 2) x = (-10 ± √(100 + 40)) / 4 x = (-10 ± √140) / 4 x = (-10 ± 2√35) / 4 x = (-5 ± √35) / 2

Therefore, the equation has two real solutions: x = (-5 + √35) / 2 and x = (-5 - √35) / 2.

Conclusion

In this guide, we have successfully demonstrated how to rewrite quadratic equations into their standard form and find their real solutions. By understanding the standard form and employing methods such as factoring, completing the square, and the quadratic formula, you can effectively solve a wide range of quadratic equations. Remember that the discriminant (b² - 4ac) provides valuable information about the nature of the roots, helping you determine whether the equation has real solutions, complex solutions, or a repeated root. Mastering these techniques is essential for building a strong foundation in mathematics and related fields.