Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(31-44\) : Solve by using the quadratic formula. $$ 4 x^{2}+81=36 x $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{9}{2} \).

Step 1: Identify the Quadratic Equation Standard Form

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). We need to rearrange the given equation \( 4x^2 + 81 = 36x \) into this form.

Step 2: Rearrange the Equation

Move all terms to one side of the equation to set it equal to zero: \[ 4x^2 - 36x + 81 = 0 \] This is now in the standard form \( ax^2 + bx + c = 0 \) with \( a = 4 \), \( b = -36 \), and \( c = 81 \).

Step 3: Write Down the Quadratic Formula

The quadratic formula to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 4: Calculate the Discriminant

The value under the square root in the quadratic formula is called the discriminant: \[ b^2 - 4ac \] Substitute \( a = 4 \), \( b = -36 \), and \( c = 81 \) into the discriminant formula: \[ (-36)^2 - 4 \times 4 \times 81 \] \[ 1296 - 1296 = 0 \] Hence, the discriminant is 0.

Step 5: Solve the Quadratic Equation

Since the discriminant is 0, we know there is one real solution. Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \[ x = \frac{-(-36) \pm \sqrt{0}}{2 \times 4} \] \[ x = \frac{36}{8} \] \[ x = \frac{9}{2} \] Thus, the solution is \( x = \frac{9}{2} \).

Key Concepts

Discriminant

The discriminant is a crucial component in solving quadratic equations using the quadratic formula. It is the part of the quadratic formula under the square root sign, expressed as \( b^2 - 4ac \). This part of the equation tells us about the nature of the roots. Here is why the discriminant is important:

• If the discriminant is greater than zero, \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If the discriminant is equal to zero, \( b^2 - 4ac = 0 \), there is exactly one real root, meaning the roots are repeated or identical.
• If the discriminant is less than zero, \( b^2 - 4ac < 0 \), the equation has no real roots but two complex roots.

In our specific exercise, the discriminant was calculated and found to be zero, \( b^2 - 4ac = 0 \), indicating that the quadratic equation has one real root. This happens because when the value of the discriminant is zero, the quadratic formula simplifies to only one solution, as the "plus/minus" part in \( \pm \sqrt{0} \) becomes zero.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \( x \), featuring a simple yet impactful mathematical form: \( ax^2 + bx + c = 0 \). When solving quadratic equations, we often use the quadratic formula, but it is equally important to understand the nature of these equations. Key points about quadratic equations include:

• They always have the term \( ax^2 \) where \( a eq 0 \), making them non-linear and parabolic in graphical representation.
• The coefficients \( a \), \( b \), and \( c \) can be any real number, dictating the position and shape of the parabola.
• The solutions or roots of the quadratic equation can be real or complex numbers, depending on the discriminant.

In the context of our problem, the quadratic equation started as \( 4x^2 + 81 = 36x \) and was rearranged to standard form \( 4x^2 - 36x + 81 = 0 \), allowing us to use the quadratic formula effectively.

Standard Form

The standard form of a quadratic equation is written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. Understanding and using this form is fundamental when solving quadratic equations, particularly when applying the quadratic formula. Here's what you need to know about the standard form:

• It is crucial for identifying the coefficients \( a \), \( b \), and \( c \), which are needed to apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Rearranging any quadratic equation into the standard form ensures all terms are on one side of the equation set to zero, making the application of the formula direct and straightforward.
• From this form, you can also determine the axis of symmetry and vertex of the graph of the equation, which are advanced topics related to the graphical analysis of quadratic functions.

In our exercise, the equation was initially given as \( 4x^2 + 81 = 36x \). By moving all terms to one side, it was transformed into the standard form \( 4x^2 - 36x + 81 = 0 \), leading us to solve it using the quadratic formula.