Solving $25x^2 - 60x + 37 = 0$: Find Roots In A+bi Form

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Let's dive into finding the roots of the quadratic equation 25x2−60x+37=025x^2 - 60x + 37 = 0 and express them in the simplest a+bia + bi form. Quadratic equations pop up everywhere in mathematics, physics, and engineering, so understanding how to solve them is super useful. This particular equation might look a bit tricky because it involves complex numbers, but don't worry, we'll break it down step by step.

Understanding Quadratic Equations

Before we tackle our specific equation, let's recap what a quadratic equation is. A quadratic equation is generally expressed in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable we want to find. The solutions to this equation are also known as the roots, and they can be real or complex numbers. The discriminant, given by the formula b2−4acb^2 - 4ac, tells us a lot about the nature of the roots. If the discriminant is positive, we have two distinct real roots. If it's zero, we have one repeated real root. And if it's negative, we have two complex conjugate roots. Complex roots always come in pairs of the form a+bia + bi and a−bia - bi, where aa and bb are real numbers, and ii is the imaginary unit, defined as i=−1i = \sqrt{-1}. Understanding the discriminant helps us anticipate the type of solutions we'll find, which is always a good starting point. Also, remember that complex numbers extend the real number system, allowing us to solve equations that have no real solutions. Complex numbers are not just abstract concepts; they have practical applications in fields like electrical engineering, where they are used to analyze alternating current circuits, and in quantum mechanics, where they are fundamental to describing the behavior of particles at the atomic level. So, mastering the techniques to solve quadratic equations with complex roots is not just an academic exercise but a valuable skill with real-world implications.

Applying the Quadratic Formula

For our equation 25x2−60x+37=025x^2 - 60x + 37 = 0, we can identify a=25a = 25, b=−60b = -60, and c=37c = 37. The quadratic formula is a powerful tool that provides a direct way to find the roots of any quadratic equation. The formula is given by: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$ Plugging in our values, we get $x = \frac{-(-60) \pm \sqrt{(-60)^2 - 4(25)(37)}2(25)}$ Now, let's simplify this expression step by step. First, we have $x = \frac{60 \pm \sqrt{3600 - 3700}50}$ This simplifies to $x = \frac{60 \pm \sqrt{-100}50}$ Since we have a negative number under the square root, we know we're dealing with complex roots. We can rewrite −100\sqrt{-100} as 100⋅−1\sqrt{100} \cdot \sqrt{-1}, which is 10i10i. So, our equation becomes $x = \frac{60 \pm 10i50}$ Now, we can simplify further by dividing both the real and imaginary parts by 10 $x = \frac{6 \pm i5}$ Finally, we can express the roots in the form a+bia + bi $x = \frac{6{5} \pm \frac{1}{5}i$ Thus, the roots are 65+15i\frac{6}{5} + \frac{1}{5}i and 65−15i\frac{6}{5} - \frac{1}{5}i. These are the two complex conjugate roots of the given quadratic equation. Remember, the ±\pm sign in the quadratic formula gives us two solutions: one with addition and one with subtraction. In the case of complex roots, these solutions are complex conjugates, meaning they have the same real part but opposite imaginary parts. The quadratic formula is not just a formula to memorize; it's a result derived from completing the square, a fundamental technique in algebra. Understanding the derivation of the quadratic formula can provide deeper insights into why it works and how it relates to other algebraic concepts.

Expressing the Roots in Simplest a + bi Form

As we found in the previous section, the roots of the equation 25x2−60x+37=025x^2 - 60x + 37 = 0 are: $x = \frac{6}{5} + \frac{1}{5}i$ and $x = \frac{6}{5} - \frac{1}{5}i$ These are already in the simplest a+bia + bi form, where a=65a = \frac{6}{5} and b=±15b = \pm \frac{1}{5}. The term "simplest form" in this context means that the real and imaginary parts are expressed as reduced fractions. In our case, 65\frac{6}{5} and 15\frac{1}{5} cannot be simplified further. Expressing complex numbers in the standard a+bia + bi form makes it easier to perform arithmetic operations such as addition, subtraction, multiplication, and division. It also allows for a clear representation of the complex number on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The complex plane provides a visual way to understand complex numbers and their relationships to each other. For example, the magnitude of a complex number a+bia + bi can be found using the Pythagorean theorem as a2+b2\sqrt{a^2 + b^2}, and the angle it makes with the positive real axis can be found using trigonometric functions. So, while expressing the roots in simplest a+bia + bi form might seem like a minor detail, it is an essential step for further analysis and application of these complex numbers.

Verification

To verify that our roots are correct, we can plug them back into the original equation 25x2−60x+37=025x^2 - 60x + 37 = 0. Let's start with x=65+15ix = \frac{6}{5} + \frac{1}{5}i: $25(\frac6}{5} + \frac{1}{5}i)^2 - 60(\frac{6}{5} + \frac{1}{5}i) + 37$ First, let's expand the square $(\frac{65} + \frac{1}{5}i)^2 = (\frac{6}{5})^2 + 2(\frac{6}{5})(\frac{1}{5}i) + (\frac{1}{5}i)^2 = \frac{36}{25} + \frac{12}{25}i - \frac{1}{25} = \frac{35}{25} + \frac{12}{25}i$ Now, plug this back into the equation $25(\frac{3525} + \frac{12}{25}i) - 60(\frac{6}{5} + \frac{1}{5}i) + 37$ Simplify $35 + 12i - 72 - 12i + 37$ Combine like terms: $(35 - 72 + 37) + (12i - 12i) = 0$ So, the first root checks out. Now let's verify the second root $x = \frac{65} - \frac{1}{5}i$ $25(\frac{65} - \frac{1}{5}i)^2 - 60(\frac{6}{5} - \frac{1}{5}i) + 37$ Expand the square $(\frac{65} - \frac{1}{5}i)^2 = (\frac{6}{5})^2 - 2(\frac{6}{5})(\frac{1}{5}i) + (\frac{1}{5}i)^2 = \frac{36}{25} - \frac{12}{25}i - \frac{1}{25} = \frac{35}{25} - \frac{12}{25}i$ Plug this back into the equation $25(\frac{35{25} - \frac{12}{25}i) - 60(\frac{6}{5} - \frac{1}{5}i) + 37$ Simplify: $35 - 12i - 72 + 12i + 37$ Combine like terms: $(35 - 72 + 37) + (-12i + 12i) = 0$ The second root also checks out. Therefore, our solutions are correct. Verification is a crucial step in problem-solving because it ensures that the solutions we have obtained are indeed correct. In the case of quadratic equations, plugging the roots back into the original equation and showing that it satisfies the equation is a solid way to confirm the accuracy of the solutions. This process not only validates the roots but also helps reinforce the understanding of the relationship between the roots and the coefficients of the quadratic equation.

Conclusion

In summary, the roots of the quadratic equation 25x2−60x+37=025x^2 - 60x + 37 = 0 in simplest a+bia + bi form are: $x = \frac{6}{5} + \frac{1}{5}i$ and $x = \frac{6}{5} - \frac{1}{5}i$ We found these roots by using the quadratic formula and simplifying the resulting expressions. We also verified our solutions by plugging them back into the original equation. Understanding how to solve quadratic equations with complex roots is a valuable skill in various fields of mathematics and science. Keep practicing, and you'll become a pro at solving these types of problems! For further reading on quadratic equations, consider checking out resources like Khan Academy's quadratic equations section.