5 Key Steps To Solve X² + 5x + 14 = 0

When faced with the quadratic equation ( x^2 + 5x + 14 = 0 ), it can initially seem daunting. However, breaking it down into manageable steps makes the process easier and more intuitive. Let’s explore the key steps to solve this equation effectively, along with some helpful tips, common mistakes to avoid, and practical examples to enhance your understanding.

Step 1: Identify the Coefficients

The first step is to identify the coefficients in your quadratic equation, which is generally in the form of ( ax^2 + bx + c = 0 ).

• In this equation, ( a = 1 ), ( b = 5 ), and ( c = 14 ).

This information is crucial for applying various methods to solve the quadratic equation.

Step 2: Calculate the Discriminant

Next, we need to calculate the discriminant (( D )), which helps us determine the nature of the roots of the quadratic equation. The formula for the discriminant is:

[ D = b^2 - 4ac ]

Substituting the values of ( a ), ( b ), and ( c ):

[ D = 5^2 - 4(1)(14) = 25 - 56 = -31 ]

Since the discriminant is negative, we know that the quadratic equation has no real solutions; instead, it has two complex (imaginary) solutions. This is a critical realization that can help direct our next steps.

Step 3: Apply the Quadratic Formula

To find the roots of the equation, we use the quadratic formula:

[ x = \frac{{-b \pm \sqrt{D}}}{{2a}} ]

Given that ( D = -31 ), substituting the values gives:

[ x = \frac{{-5 \pm \sqrt{-31}}}{{2 \cdot 1}} ]

This leads us to a realization: since we're dealing with a negative square root, we can express the roots in terms of imaginary numbers. We can rewrite ( \sqrt{-31} ) as ( i\sqrt{31} ), where ( i ) is the imaginary unit.

Thus, we have:

[ x = \frac{{-5 \pm i\sqrt{31}}}{2} ]

This results in two complex solutions:

[ x_1 = \frac{{-5 + i\sqrt{31}}}{2} ] [ x_2 = \frac{{-5 - i\sqrt{31}}}{2} ]

Step 4: Expressing the Solutions

Now that we have found our solutions, it's often helpful to express them in a cleaner format. The solutions can be written as:

• ( x_1 = -\frac{5}{2} + \frac{\sqrt{31}}{2}i )
• ( x_2 = -\frac{5}{2} - \frac{\sqrt{31}}{2}i )

These represent the two complex roots of the original equation ( x^2 + 5x + 14 = 0 ).

Step 5: Validate the Solutions (Optional)

While not always necessary, it’s a good practice to validate your solutions by substituting them back into the original equation to ensure that they satisfy the equation.

You can substitute ( x_1 ) and ( x_2 ) back into ( x^2 + 5x + 14 ) and confirm they equate to zero. Due to the complexity, this is a computational step that can often be skipped but can help reinforce your understanding of the solutions.

Common Mistakes to Avoid

• Neglecting the Discriminant: Always check the discriminant first! It tells you whether to expect real or complex solutions.
• Ignoring Complex Numbers: If you encounter a negative discriminant, remember to use ( i ) (the imaginary unit) correctly.
• Misapplying the Quadratic Formula: Double-check your signs and calculations while applying the formula.

Troubleshooting Tips

If you find yourself confused at any point, try the following:

• Revisit the Coefficients: Ensure that you've correctly identified ( a ), ( b ), and ( c ).
• Check Your Arithmetic: Mistakes can often be traced back to simple arithmetic errors in calculating the discriminant or substituting values into the quadratic formula.
• Visualize: Graphing the quadratic can provide insight into the nature of its roots. If it doesn't cross the x-axis, you likely have complex solutions.

<div class="faq-section"> <div class="faq-container"> <h2>Frequently Asked Questions</h2> <div class="faq-item"> <div class="faq-question"> <h3>Can all quadratic equations have real roots?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>No, quadratic equations can have real roots, complex roots, or repeated roots, depending on the discriminant value.</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>What does a negative discriminant mean?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>A negative discriminant indicates that the quadratic equation has two complex solutions (no real roots).</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>How do I verify my solutions?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Substitute your solutions back into the original quadratic equation to see if they satisfy it (i.e., equal zero).</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>What are complex roots?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Complex roots are solutions that involve imaginary numbers, often resulting from a negative discriminant.</p> </div> </div> </div> </div>

In summary, solving the quadratic equation ( x^2 + 5x + 14 = 0 ) is a straightforward process when you break it down into steps. From identifying coefficients to applying the quadratic formula, each stage builds on the last. Remember to look out for the discriminant; it’s your first clue regarding the types of solutions you’ll find.

Practice these steps with other quadratic equations, and soon you’ll become proficient in tackling even the trickiest problems. Happy solving!

<p class="pro-note">🔑Pro Tip: Always double-check your calculations and familiarize yourself with complex numbers to enhance your problem-solving skills.</p>