If the equation \(x^{2}-m(2 x-8)-15=0\) has equal roots then \(\mathrm{m}=\) (a) \(3,-5\) (b) \(-3,5\) (c) 3,5 (d) \(-3,-5\)

Understanding the discriminant is crucial when solving quadratic equations, as it gives us valuable information about the nature of the roots. In a standard quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant, denoted by \(D\), is calculated using the formula \(D = b^2 - 4ac\).

The value of the discriminant determines the number and type of the roots:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, also known as equal roots or a 'double root'.
• If \(D < 0\), the equation has two complex roots, which are conjugates of each other.

In the provided exercise, we were interested in the scenario when the quadratic equation has equal roots. This occurs when the discriminant is zero, \(D = 0\). By applying this condition to the exercise equation and setting up an equation in terms of \(m\), we can solve for the value(s) of \(m\) that result in equal roots.

When solving quadratic equations, one effective method is factoring. Factoring involves expressing the quadratic equation \(ax^2 + bx + c = 0\) as a product of two binomials. This method is most efficient when the equation is factorable over the integers, meaning it can be written in the form \((dx + e)(fx + g) = 0\), where \(d, e, f,\) and \(g\) are integers.

To factor successfully:

• Seek two numbers that multiply to give \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and add to give \(b\) (the coefficient of \(x\)).
• Once the two numbers are found, the quadratic equation can be expressed as a product of two binomials using these numbers.
• Set each binomial containing \(x\) equal to zero and solve for \(x\) to find the roots of the equation.

In the accompanying exercise, we factored the equation \(m^2 - 8m + 15 = 0\) into \((m - 3)(m - 5) = 0\). By setting each factor equal to zero, we determined the values of \(m\) that give the quadratic equation equal roots.

There are several methods to solve quadratic equations, each with its own use-cases and advantages. The fundamental goal is to find the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).

The three main methods are:

• Factoring the equation, as previously mentioned.
• Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square, which involves rewriting the equation in the form \((x - p)^2 = q\) to then find the value of \(x\).

The choice of method often depends on the coefficients of the equation and the ease of application. For example, factoring is often the simplest route when the equation is factorable with integer coefficients, while the quadratic formula is a reliable all-purpose tool, especially when dealing with complex coefficients or when the equation is not easily factored. In the present exercise, after determining the discriminant to be zero, we factored the quadratic equation and found the values of \(m\), hence solving the quadratic equation for equal roots.