Question

The number of real roots of the equation \((x-1)^{2}+(x-2)^{2}+(x-3)^{2}=0\) is (a) 2 (b) 1 (c) 0 (d) 3

Short answer

Answer: (c) 0 real roots

Step-by-step solution

Expand the Terms

Let's expand the terms on the left-hand side of the equation: \((x-1)^{2}+(x-2)^{2}+(x-3)^{2}\). Expanding these terms, we get: \((x-1)^{2} = x^2 - 2x + 1\) \((x-2)^{2} = x^2 - 4x + 4\) \((x-3)^{2} = x^2 - 6x + 9\)

Sum the Expanded Terms

Now we will sum these three expanded terms: \((x^2 - 2x + 1) + (x^2 - 4x + 4) + (x^2 - 6x + 9) = 0\) Combining like terms, we get: \(3x^2 - 12x + 14 = 0\)

Analyze the Resulting Equation

We now have the quadratic equation \(3x^2 - 12x + 14 = 0\). To determine the number of real roots for this equation, we need to calculate its discriminant and check its value: \(\Delta = b^2 - 4ac\), where \(a = 3\), \(b = -12\), and \(c = 14\). Using these values, we calculate the discriminant: \(\Delta = (-12)^2 - 4(3)(14) = 144 - 168 = -24\) Since the discriminant is negative, we can conclude that the quadratic equation has no real roots. Therefore, the correct answer is (c) 0 real roots.

Key concepts

Quadratic Equations

At the heart of understanding algebra lies the concept of quadratic equations. A quadratic equation is typically in the form of \(ax^2+bx+c=0\) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are vital because they appear in various branches of mathematics and even in real-world scenarios such as calculating areas, determining object trajectories, and optimizing profit in business applications.

Quadratic equations are unique because they can have up to two real solutions. These solutions are the values of x that make the equation true, and they are commonly referred to as the 'roots' of the equation. Understanding how to find these roots is crucial for solving quadratic problems efficiently.

The nature of the roots – whether they are real or complex – is determined by a component called the 'discriminant'. This brings us to ask: what can the discriminant tell us about the roots of a quadratic equation? This question leads to a deeper analysis of the discriminant's role in our next section.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation, represented by the Greek letter Delta (\( \Delta \)), offers insight into the nature and number of roots of the equation without actually solving it. To find the discriminant, we use the formula \( \Delta = b^2 - 4ac \), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

The value of the discriminant can reveal three possible scenarios for the roots of the quadratic equation:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (also known as a repeated or double root).
• If \(\Delta < 0\), there are no real roots, but rather a pair of complex conjugate roots.

In the exercise provided, the discriminant was found to be negative. This indicates that our quadratic equation has no real solutions but instead has complex roots. Hence, the number of real roots for the equation \((x-1)^2+(x-2)^2+(x-3)^2=0\) is zero, corresponding to our choice (c) 0.

Expanding Polynomial Expressions

Expanding polynomial expressions is a foundational skill in algebra, essential for simplifying and solving equations. When faced with the problem of determining the number of real roots, we first need to expand our expressions to transform the equation into a recognizable quadratic form. Expanding involves applying algebraic identities, such as the square of a binomial \((x - a)^2 = x^2 - 2ax + a^2\), and combining like terms.

Through expansion, complex expressions are made more manageable, leading to equations that are easier to analyze. In the initial exercise, for instance, we expanded three squared binomial terms and combined them thereafter, ending up with a single quadratic equation. We then applied our knowledge about discriminants to this new form to determine the number of real roots. It is crucial to master the technique of expanding polynomial expressions, as it enables us to tackle more complex algebraic problems and pave the way for applying further analysis such as finding the discriminant or factoring.