Solving Quadratic Equations: X<Sup>2</Sup> + 6X + 7 = 0

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Introduction

Quadratic equations are equations that contain a variable raised to the second power, such as x². These equations can be solved using a variety of methods, but one common approach is to use the quadratic formula. In this article, we will explore how to solve the quadratic equation x² + 6x + 7 = 0 using the quadratic formula.

The Quadratic Formula

The quadratic formula is a formula that can be used to solve any quadratic equation of the form ax² + bx + c = 0, where a, b, and c are constants. The formula is: x = (-b ± √(b² - 4ac)) / 2a This formula gives us the values of x that satisfy the equation. The ± symbol means that we need to consider both the positive and negative roots.

Solving x² + 6x + 7 = 0

To use the quadratic formula to solve x² + 6x + 7 = 0, we first need to identify the values of a, b, and c. In this equation, a = 1, b = 6, and c = 7. We can then plug these values into the quadratic formula: x = (-6 ± √(6² - 4(1)(7))) / 2(1) Simplifying this expression gives us: x = (-6 ± √(36 - 28)) / 2 x = (-6 ± √8) / 2

Simplifying the Square Root

We can simplify the square root in the numerator by factoring out a 4: x = (-6 ± 2√2) / 2 We can then simplify the fraction by dividing both the numerator and denominator by 2: x = -3 ± √2

The Roots of the Equation

Therefore, the solutions to the equation x² + 6x + 7 = 0 are: x = -3 + √2 or x = -3 - √2 These are the roots of the equation, which are the values of x that make the equation true.

Checking the Solutions

We can check our solutions by plugging them back into the original equation and verifying that they make the equation true. For example, if we plug in x = -3 + √2, we get: (-3 + √2)² + 6(-3 + √2) + 7 = 0 Simplifying this expression gives us: 2 - 6√2 + 9 - 18 + 6√2 + 7 = 0 This simplifies to: 18 = 18 Since the equation is true, we know that our solution is correct.

Conclusion

In this article, we explored how to solve the quadratic equation x² + 6x + 7 = 0 using the quadratic formula. We identified the values of a, b, and c, plugged them into the formula, and simplified to get the roots of the equation. We then verified our solutions by plugging them back into the original equation. Solving quadratic equations can be challenging, but with the quadratic formula, we have a reliable method for finding the solutions.