Question

The equation \(9 \mathrm{y}^{2}(\mathrm{~m}+3)+6(\mathrm{~m}-3) \mathrm{y}+(\mathrm{m}+3)=0\), where \(\mathrm{m}\) is real, has real roots. Which of the following is true? (1) \(\mathrm{m}=0\) (2) \(\mathrm{m} \leq 0\) (3) Either (1) or (2) (4) Neither (1) nor (2)

Short answer

Answer: (2) m <= 0

Step-by-step solution

Identify the coefficients of the quadratic equation

The given quadratic equation is: \(9y^2(my+3)+6(my-3)y+(m+3)=0\). Comparing it with the standard quadratic equation form, \(ay^2+by+c=0\), we can identify the coefficients as follows: - a = 9(my + 3) - b = 6(my - 3) - c = m + 3

Calculate the discriminant

The discriminant (D) of a quadratic equation is given by the formula: \(D = b^2 - 4ac\). Using the values of a, b and c we found in the previous step, let's compute the discriminant: \(D = (6(my - 3))^2 - 4(9(my + 3))(m + 3)\)

Simplify the discriminant expression

Simplify the discriminant expression: \(D = 36(m^2y^2 - 6my + 9) - 36(9m^2y^2 + 3m^2y - 27my + 27)\)

Factor out common terms

Factor out the common term 36 from the expression and simplify: \(D = 36(m^2y^2 - 6my + 9 - 9m^2y^2 - 3m^2y + 27my - 27)\) \(D = 36(-8m^2y^2 + 21my - 18)\)

Find conditions for real roots

For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero: \(D \geq 0\) \(36(-8m^2y^2 + 21my - 18) \geq 0\) Since 36 is always positive, we can divide both sides by 36: \(-8m^2y^2 + 21my - 18 \geq 0\) Now we need to find the range of values for m that satisfy this inequality.

Analyze the given options

Let's analyze the options one by one: (1) m = 0: Substitute m = 0 in the inequality: \(-8(0)^2y^2 + 21(0)y - 18 \geq 0\) \(-18 \geq 0\). This is not true, so option (1) is incorrect. (2) m <= 0: We need to find out if the inequality holds true for any negative value of m or not. Let's consider m = -1 and see if the inequality holds true: \(-8(-1)^2y^2 + 21(-1)y - 18 \geq 0\) \(-8y^2 - 21y - 18 \geq 0\) Factoring the left side, we have: \((2y + 3)(4y + 6) \geq 0\) As both factors on the left side are negative for any value of y, the inequality always holds true for any negative value of m. So, option (2) is correct. (3) Either (1) or (2): Since option (1) is incorrect, option (3) is also incorrect. (4) Neither (1) nor (2): As option (2) is correct, option (4) cannot be true. So, the correct answer is option (2).

Key concepts

Discriminant of Quadratic Equation

Understanding the discriminant is crucial when analyzing quadratic equations. In a standard quadratic equation represented by
\[ ax^2 + bx + c = 0 \],
the discriminant is found using the formula
\[ D = b^2 - 4ac \].
This single value determines the nature of the roots the equation will have. If the discriminant is positive, there will be two distinct real roots. If it is zero, there will be exactly one real root, which is also known as a repeated or double root. Lastly, if the discriminant is negative, the roots will be complex and not real numbers.
In the exercise, by calculating and simplifying the discriminant, students can infer the conditions necessary for the given equation to have real roots. The step-by-step solution provided walks through this process, emphasizing the fact that for real roots, we require that
\[ D \geq 0 \].
This forms the basis for further analysis, where the value of variable 'm' is determined to satisfy the condition for real roots.

Solving Inequalities

Solving inequalities is a process quite similar to solving equations, with the primary difference being the solution: instead of exact values, we often find ranges or intervals. When given an inequality like
\[ \textrm{Expression} \geq 0 \]
or
\[ \textrm{Expression} \leq 0 \],
our goal is to determine for which variable values the inequality holds true. This can involve various methods, such as factoring, graphing, or using the sign analysis method.
In the context of this exercise, once the discriminant is simplified and represented as an inequality, the question seeks to identify for which values of 'm' the inequality becomes true. Understanding how to manipulate and solve these inequalities is fundamental in determining the conditions that govern the nature of the roots for the quadratic equation.

Quadratic Equations

Quadratic equations form a foundational element of algebra and are represented by an equation of the form
\[ ax^2 + bx + c = 0 \],
where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The roots of a quadratic equation are the values of 'x' that satisfy the equation. These can be found by various methods: factoring, completing the square, using the quadratic formula, or graphing. The quadratic formula derived from completing the square of a quadratic equation is given by,
\[ x = \frac{{-b \pm \sqrt{{D}}}}{{2a}} \],
where \( \sqrt{{D}} \) is the square root of the discriminant.
In the given exercise, the quadratic equation takes a slightly more complex form due to the presence of the variable 'm'. However, by applying the standard techniques of solving quadratic equations and understanding the discriminant, students can evaluate the necessary conditions for 'm' to ensure the roots are real. This exercise serves as a practical application of quadratic equation concepts, challenging students to delve into variable analysis.