Question

The quadratic equation having rational coefficients and one of the roots as \(4+\sqrt{15}\) is (1) \(\mathrm{x}^{2}-8 \mathrm{x}+1=0\). (2) \(\mathrm{x}^{2}+\mathrm{x}-8=0 .\) (3) \(\mathrm{x}^{2}-\mathrm{x}+8=0\). (4) \(\mathrm{x}^{2}+8 \mathrm{x}+8=0\).

Short answer

Answer: The correct quadratic equation is \(x^{2} - 8x + 1 = 0\).

Step-by-step solution

Find the conjugate root

Given that one of the roots is \(4+\sqrt{15}\), the other root is the conjugate root which is \(4 - \sqrt{15}\).

Form quadratic equation using roots

Using the sum and product of roots, we can form the quadratic equation as follows: The sum of the roots, \(s = \alpha + \beta = (4+\sqrt{15}) + (4-\sqrt{15}) = 8\). The product of the roots, \(p = \alpha \cdot \beta = (4+\sqrt{15})(4-\sqrt{15})\). To find the product, using the difference of squares formula, \(p = (4+\sqrt{15})(4-\sqrt{15})= 4^{2} - (\sqrt{15})^{2} = 16 - 15 = 1\). Thus, the quadratic equation is \(x^{2}- s x + p =0\), which simplifies to \(x^{2} - 8x + 1 = 0\).

Match the equation obtained to the given options

Comparing the quadratic equation obtained with the options given, we find that: (1) \(x^{2}-8 x+1=0\) (Matches) (2) \(x^{2}+x-8=0\) (Does not match) (3) \(x^{2}-x+8=0\) (Does not match) (4) \(x^{2}+8 x+8=0\) (Does not match) Hence, the correct quadratic equation is (1) \(x^{2}-8 x+1=0\).

Key concepts

Conjugate Roots

When working with quadratic equations, understanding the concept of conjugate roots is crucial. These are a pair of roots in which one is the 'mirror image' of the other around the real axis of the complex plane. More simply, if one root of a quadratic equation with real coefficients is a complex number, its conjugate will also be a root of the same equation. The conjugate of a complex number is formed by changing the sign of the imaginary part, which means if one root is expressed as \((a + bi)\), then the conjugate root will be \((a - bi)\), where \(a\) is the real part and \(b\) is the imaginary part.

Applying this to real numbers with square roots: if you have a root with a real number plus a square root, like \(4 + \text{sqrt}{15}\), the conjugate root would be its pair with a minus instead of plus, which is \(4 - \text{sqrt}{15}\). This pair of roots is not only symmetrical in form but also essential to determine the coefficients of the quadratic equation they belong to.

Sum and Product of Roots

The sum and product of roots of a quadratic equation are linked to its coefficients in a very straightforward manner, which can greatly simplify the process of finding an equation from its roots. For the standard quadratic equation \(ax^{2}+bx+c=0\), where \(a\), \(b\), and \(c\) are known coefficients, the sum of the roots \((\alpha + \beta)\) is equal to \(-b/a\) and the product \((\alpha \cdot \beta)\) is equal to \(c/a\).

Let's break this down with an example. We know one root is \(4 + \text{sqrt}{15}\), therefore, to find the sum, you would simply add this root to its conjugate, that is \(4 - \text{sqrt}{15}\). Doing this, the irrational parts cancel out and you're left with the sum of the roots, which is 8. In terms of the product, when you multiply these roots together, the difference of squares applies and the result is 1. Using these values, the equation can be reconstructed as \(x^2 - sx + p = 0\), with 's' being the sum and 'p' the product of the roots.

Difference of Squares

The difference of squares is a helpful algebraic pattern that occurs when you have the difference between two squared numbers, typically presented as \(a^{2} - b^{2}\). This pattern can be factored into \((a + b)(a - b)\). It is particularly useful in simplifying expressions and is a key player in understanding the product of conjugate roots.

For instance, when we calculated the product of the roots \((4 + \text{sqrt}{15})\) and \((4 - \text{sqrt}{15})\), we effectively used the difference of squares formula, which leads to \(4^{2} - (\text{sqrt}{15})^{2}\). After performing the subtraction, \(16 - 15\), we found the product to be simply 1. This principle is fundamental not only in simplifying but also in solving equations and understanding the relationship between algebraic expressions and their geometric representations.