Problem 34 - Basic Technical Mathematics with Calculus

Question

Solve the given applied problem. Find the smallest integral value of \(c\) such that \(y=3 x^{2}-12 x+c\) has no real roots.

Step-by-Step Solution

Answer

The smallest integral value of \(c\) is 13.

Step 1: Understanding the Discriminant

In a quadratic equation of the form \[ ax^2 + bx + c = 0, \]the discriminant is given by \[ D = b^2 - 4ac. \]For a quadratic equation to have no real roots, its discriminant must be less than zero (\(D < 0\)). For the given equation, identify the values of \(a\), \(b\), and \(c\).

Step 2: Identify Coefficients

The given quadratic equation is \[ y = 3x^2 - 12x + c. \]Here, \(a = 3\), \(b = -12\), and \(c\).

Step 3: Calculate the Discriminant

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \[ D = (-12)^2 - 4(3)(c) = 144 - 12c. \]

Step 4: Set the Condition for No Real Roots

For the equation to have no real roots, set the discriminant less than zero:\[ 144 - 12c < 0. \]

Step 5: Solve the Inequality for c

Solve the inequality \[ 144 - 12c < 0 \] for \(c\):1. Subtract 144 from both sides: \[ -12c < -144. \]2. Divide by -12 and reverse the inequality sign: \[ c > 12. \]

Step 6: Determine the Smallest Integer Value

The smallest integer greater than 12 is 13. Therefore, the smallest integral value of \(c\) such that the quadratic equation has no real roots is 13.

Key Concepts

Understanding the Discriminant

When dealing with quadratic equations, the discriminant is crucial in determining how many real roots, if any, the equation possesses. For an equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula \( D = b^2 - 4ac \). The value of \( D \) guides us in understanding the nature 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 a repeated root.
• If \( D < 0 \), the equation has no real roots and its roots are complex or imaginary.

Thus, to ensure a quadratic equation has no real roots, we must find values for \( c \) such that \( D < 0 \). This involves manipulating the coefficients of the equation.

Finding Real Roots

Real roots are the solutions to the quadratic equation where the graph of the equation crosses or touches the x-axis. These occur when the discriminant is zero or positive. In other situations (\( D < 0 \)), the parabola does not intersect the x-axis at any point.

• The x-values where these crossings occur are the real roots of the equation.
• An equation can have zero, one, or two real roots depending on the discriminant value.

Finding real roots graphically involves looking at the interception points on the x-axis, while computationally it involves solving the quadratic formula or factoring, if possible. Avoiding real roots involves setting conditions on the discriminant so it remains negative for a given set of coefficients.

Solving Inequalities in Algebra

An inequality describes a relationship of non-equality between two expressions. In the case of finding the smallest integral value of \( c \) for which the quadratic equation has no real roots, an inequality guides us in determining the valid range of coefficients.

• To solve the inequality \( 144 - 12c < 0 \), begin by rearranging terms to isolate the variable \( c \).
• Subtract 144 from both sides: \( -12c < -144 \).
• Divide each side by -12, remembering to flip the inequality sign, resulting in \( c > 12 \).

This tells us that for \( c \) to cause the discriminant to be less than zero, it must be more than 12. Thus, when searching for the smallest integer satisfying this, we identify \( c = 13 \) as it meets \( c > 12 \). Solving inequalities is essential in finding conditions on variables in algebraic problems.

Coefficients in Algebra

Coefficients are the numbers before variables in algebraic expressions and they play a key role in determining the characteristics of quadratic equations. In our equation, \( y = 3x^2 - 12x + c \), we identify:

• \( a = 3 \), which multiplies the \( x^2 \) term.
• \( b = -12 \), which multiplies the \( x \) term.
• \( c \) is the constant term.

The coefficient \( a \) influences the direction and width of the parabola's graph. A positive \( a \) means the parabola opens upwards, while a negative \( a \) indicates it opens downwards. The coefficient \( b \) affects the position of the vertex along the x-axis and the value of \( c \) determines the vertical shift of the parabola. Each coefficient's sign and magnitude deeply impact the shape and position of the quadratic's graph. Understanding coefficients helps in visualizing and solving quadratic equations efficiently, especially when adjusting them to change root conditions.