Solving $x^2 + 2x + 1 = 17$ A Step By Step Guide

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When faced with a quadratic equation such as x2+2x+1=17x^2 + 2x + 1 = 17, the task is to find the values of xx that satisfy the equation. Quadratic equations, characterized by the highest power of the variable being 2, often present multiple solutions. Several methods can be employed to solve them, including factoring, completing the square, and using the quadratic formula. In this comprehensive guide, we'll explore the step-by-step solution to the given equation and delve into the underlying principles of quadratic equation solving.

Understanding the Equation

At its core, the given equation x2+2x+1=17x^2 + 2x + 1 = 17 is a polynomial equation of degree 2, making it a quadratic equation. The standard form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and aa is not equal to zero. The solutions to this equation, also known as roots or zeros, represent the values of xx that make the equation true.

Identifying the components of the equation is the first step. In our case, we have x2+2x+1=17x^2 + 2x + 1 = 17. To bring it into the standard form, we need to subtract 17 from both sides of the equation:

x2+2x+1−17=0x^2 + 2x + 1 - 17 = 0

Which simplifies to:

x2+2x−16=0x^2 + 2x - 16 = 0

Now we can clearly see that a=1a = 1, b=2b = 2, and c=−16c = -16.

Method 1: Completing the Square

Completing the square is a powerful technique to transform a quadratic equation into a perfect square trinomial, making it easier to solve. The idea is to manipulate the equation so that one side becomes a squared expression.

Starting with our equation x2+2x−16=0x^2 + 2x - 16 = 0, we focus on the terms involving xx, namely x2+2xx^2 + 2x. To complete the square, we need to add and subtract a value that makes the expression a perfect square. This value is determined by taking half of the coefficient of the xx term (which is 2), squaring it (which gives us (2/2)2=1(2/2)^2 = 1), and adding and subtracting it within the equation:

x2+2x+1−1−16=0x^2 + 2x + 1 - 1 - 16 = 0

Now, we can rewrite the first three terms as a perfect square:

(x+1)2−1−16=0(x + 1)^2 - 1 - 16 = 0

Simplifying further, we get:

(x+1)2−17=0(x + 1)^2 - 17 = 0

Now, isolate the squared term by adding 17 to both sides:

(x+1)2=17(x + 1)^2 = 17

To eliminate the square, we take the square root of both sides. Remember to consider both positive and negative roots:

x+1=±17x + 1 = \pm \sqrt{17}

Finally, solve for xx by subtracting 1 from both sides:

x=−1±17x = -1 \pm \sqrt{17}

Therefore, the solutions are x=−1+17x = -1 + \sqrt{17} and x=−1−17x = -1 - \sqrt{17}.

Method 2: The Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It provides a direct method for finding the roots, regardless of whether the equation can be easily factored or completed the square. The formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our equation, x2+2x−16=0x^2 + 2x - 16 = 0, we have a=1a = 1, b=2b = 2, and c=−16c = -16. Substitute these values into the quadratic formula:

x=−2±22−4(1)(−16)2(1)x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-16)}}{2(1)}

Simplify the expression step by step:

x=−2±4+642x = \frac{-2 \pm \sqrt{4 + 64}}{2}

x=−2±682x = \frac{-2 \pm \sqrt{68}}{2}

Now, simplify the square root. Since 68=4×1768 = 4 \times 17, we have 68=4×17=217\sqrt{68} = \sqrt{4 \times 17} = 2\sqrt{17}:

x=−2±2172x = \frac{-2 \pm 2\sqrt{17}}{2}

Divide both terms in the numerator by 2:

x=−1±17x = -1 \pm \sqrt{17}

Again, we arrive at the same solutions: x=−1+17x = -1 + \sqrt{17} and x=−1−17x = -1 - \sqrt{17}. This confirms the correctness of our solution.

Detailed Explanation of the Quadratic Formula Derivation

The quadratic formula is not just a magical formula that appears out of nowhere. It is derived from the method of completing the square applied to the general quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. This derivation showcases the power of algebraic manipulation and provides a deeper understanding of why the formula works.

  1. Start with the general form: ax2+bx+c=0ax^2 + bx + c = 0
  2. Divide by 'a': Assuming a≠0a \neq 0, divide the entire equation by aa to make the coefficient of x2x^2 equal to 1: x2+bax+ca=0x^2 + \frac{b}{a}x + \frac{c}{a} = 0
  3. Move the constant term: Subtract ca\frac{c}{a} from both sides to isolate the xx terms: x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}
  4. Complete the square: Take half of the coefficient of xx (which is b2a\frac{b}{2a}), square it (which gives (b2a)2=b24a2(\frac{b}{2a})^2 = \frac{b^2}{4a^2}), and add it to both sides of the equation. This step creates a perfect square trinomial on the left side: x2+bax+b24a2=−ca+b24a2x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}
  5. Rewrite as a squared expression: The left side is now a perfect square, which can be written as: (x+b2a)2=−ca+b24a2(x + \frac{b}{2a})^2 = -\frac{c}{a} + \frac{b^2}{4a^2}
  6. Simplify the right side: Find a common denominator and combine the terms on the right side: (x+b2a)2=b2−4ac4a2(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}
  7. Take the square root: Take the square root of both sides, remembering to include both positive and negative roots: x+b2a=±b2−4ac4a2x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}
  8. Simplify the square root: Simplify the square root on the right side: x+b2a=±b2−4ac2ax + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}
  9. Isolate x: Subtract b2a\frac{b}{2a} from both sides to solve for xx: x=−b2a±b2−4ac2ax = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
  10. Combine terms: Combine the fractions to get the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This derivation highlights how the quadratic formula is a direct result of applying the completing the square method to the general quadratic equation. Understanding this derivation not only reinforces the formula but also provides insight into the underlying mathematical principles.

Verifying the Solutions

Verification is a crucial step in the problem-solving process. It ensures that the solutions obtained are correct and satisfy the original equation. To verify our solutions, x=−1+17x = -1 + \sqrt{17} and x=−1−17x = -1 - \sqrt{17}, we substitute each value back into the original equation x2+2x+1=17x^2 + 2x + 1 = 17.

Verification for x=−1+17x = -1 + \sqrt{17}

Substitute x=−1+17x = -1 + \sqrt{17} into the equation:

(−1+17)2+2(−1+17)+1=17(-1 + \sqrt{17})^2 + 2(-1 + \sqrt{17}) + 1 = 17

Expand the squared term:

(1−217+17)+(−2+217)+1=17(1 - 2\sqrt{17} + 17) + (-2 + 2\sqrt{17}) + 1 = 17

Combine the terms:

1−217+17−2+217+1=171 - 2\sqrt{17} + 17 - 2 + 2\sqrt{17} + 1 = 17

Notice that the −217-2\sqrt{17} and 2172\sqrt{17} terms cancel each other out:

1+17−2+1=171 + 17 - 2 + 1 = 17

17=1717 = 17

Since the equation holds true, x=−1+17x = -1 + \sqrt{17} is indeed a solution.

Verification for x=−1−17x = -1 - \sqrt{17}

Substitute x=−1−17x = -1 - \sqrt{17} into the equation:

(−1−17)2+2(−1−17)+1=17(-1 - \sqrt{17})^2 + 2(-1 - \sqrt{17}) + 1 = 17

Expand the squared term:

(1+217+17)+(−2−217)+1=17(1 + 2\sqrt{17} + 17) + (-2 - 2\sqrt{17}) + 1 = 17

Combine the terms:

1+217+17−2−217+1=171 + 2\sqrt{17} + 17 - 2 - 2\sqrt{17} + 1 = 17

Again, the 2172\sqrt{17} and −217-2\sqrt{17} terms cancel each other out:

1+17−2+1=171 + 17 - 2 + 1 = 17

17=1717 = 17

Since the equation holds true, x=−1−17x = -1 - \sqrt{17} is also a solution.

Both solutions satisfy the original equation, which confirms the correctness of our results.

Conclusion

In summary, solving the quadratic equation x2+2x+1=17x^2 + 2x + 1 = 17 involves several steps, but the key is to bring the equation into a manageable form. We explored two methods: completing the square and using the quadratic formula. Both methods led us to the solutions x=−1+17x = -1 + \sqrt{17} and x=−1−17x = -1 - \sqrt{17}. Additionally, we emphasized the importance of verifying the solutions by substituting them back into the original equation. This ensures the accuracy of our work and enhances our understanding of quadratic equations.

Whether you choose to complete the square or apply the quadratic formula, understanding the underlying principles is crucial for mastering quadratic equation solving. These skills are fundamental in various areas of mathematics and its applications, making the effort to understand them well worthwhile.

The correct answer is A. x=−1±17x = -1 \pm \sqrt{17}.