Question

Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. \(24 x=-48-3 x^2\)

Short answer

The discriminant of the given quadratic equation is 0, and thus, it has exactly one real solution.

Step-by-step solution

Convert to Standard Form

The given equation is \(24x = -48 -3x^2\). Rewrite this equation to the standard form, we get \(3x^2 + 24x + 48 = 0.\)

Identify Coefficients

From the standard form, identify the coefficients \(a, b, c\) of the equation. Here, \(a = 3\), \(b = 24\) and \(c=48\).

Calculate the Discriminant

The discriminant, \(D\), of a quadratic equation \(ax^2 + bx + c=0\) is \(D = b^2 - 4ac\). Substituting \(a = 3\), \(b = 24\), and \(c = 48\) into this formula we get \(D = (24)^2 - 4*3*48 = 576 - 576 = 0.\)

Evaluate the Nature of the Roots

Since the value of the discriminant (D) is zero, it indicates that the given quadratic equation has exactly one real solution.

Key concepts

Standard Form of Quadratic Equation

Understanding the standard form of a quadratic equation is the cornerstone of unraveling its properties and predicting the nature of its solutions. A quadratic equation is typically written in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are coefficients. Crucially, \( a \), the coefficient of the squared term, must not be zero, for if it were, the equation would cease to be quadratic and become linear.
By using this agreed-upon form, we can effectively compare and analyze different quadratic equations, applying uniform methodologies to solve them. In the provided exercise, for instance, we took the equation \( 24x = -48 - 3x^2 \) and rearranged it to \( 3x^2 + 24x + 48 = 0 \), aligning it with the standard form and laying the groundwork for further exploration into its solutions.

Nature of Quadratic Roots

Deciphering the nature of the roots of a quadratic equation is like a detective decoding clues to solve a mystery. The discriminant of the quadratic formula provides us with a critical hint. \( D = b^2 - 4ac \) reveals whether the roots are real or complex, and if real, whether they are distinct or the same. If the discriminant is positive, we find two distinct real roots. A discriminant of zero indicates a single repeated real root. Conversely, a negative discriminant is a tell-tale sign of non-real complex roots.
In practical terms, if you throw a ball, plotting its path will yield a parabolic curve, analogous to the graph of a quadratic equation. The points where this curve touches the horizontal ground (x-axis) are the real roots. No touch signifies complex roots, while a single touchpoint indicates that the ball bounced off the ground at the plot of the repeated root.

Calculating Discriminant

To elucidate on the calculating discriminant, let's dive right into the process. It is computed using the formula \( D = b^2 - 4ac \), derived from the coefficients of the quadratic equation in standard form. This computation forms the basis for not only determining the nature of the quadratic roots but also for choosing the appropriate method to find them.

Delving into Discriminants

By substituting the values from the example \( a = 3 \) (coefficient of \( x^2 \) term), \( b = 24 \) (coefficient of \( x \) term), and \( c = 48 \) (constant term) into the discriminant formula, we find that \( D = 576 - 576 = 0 \). The zero discriminant concisely indicates that the quadratic equation possesses a singular, unique real solution. Variations in the values of \( a \) , \( b \) , and \( c \) would lead to different discriminant values, hence different narratives about the quadratic equation's roots.