Problem 72 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$8(4 y 2+3)-3(28 y-1)=0$$

Step-by-Step Solution

Answer

The solutions are \(y = \frac{9}{4}\) and \(y = \frac{3}{8}\).

Step 1: Expand the expression

Start by expanding the given equation: \(8(4y^2 + 3) - 3(28y - 1) = 0\). Distribute the multiplication over addition inside the parentheses.\[8 \times 4y^2 + 8 \times 3 - 3 \times 28y + 3 \times 1 = 0\] This simplifies to:\[32y^2 + 24 - 84y + 3 = 0\] Combine the constant terms:\[32y^2 - 84y + 27 = 0\]

Step 2: Identify coefficients

Ensure the equation is in the standard form \(ax^2 + bx + c = 0\). Here, \(a = 32\), \(b = -84\), \(c = 27\).

Step 3: Write the quadratic formula

The quadratic formula is given by:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 4: Calculate the discriminant

The discriminant \(D\) is calculated using the formula \(b^2 - 4ac\).\[D = (-84)^2 - 4 \times 32 \times 27\]\[D = 7056 - 3456\]\[D = 3600\]

Step 5: Solve using the quadratic formula

Substitute \(b\), \(a\), and \(c\) into the quadratic formula:\[y = \frac{-(-84) \pm \sqrt{3600}}{2 \times 32}\]\[y = \frac{84 \pm 60}{64}\]Calculate the two possible solutions for \(y\):1. \(y_1 = \frac{84 + 60}{64} = \frac{144}{64} = \frac{9}{4}\)2. \(y_2 = \frac{84 - 60}{64} = \frac{24}{64} = \frac{3}{8}\)

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. This means its highest exponent is two. The standard form of a quadratic equation is given by the formula:
\[ ax^2 + bx + c = 0 \]
where:

• \(a\), \(b\), and \(c\) are constants
• \(x\) represents the unknown variable.

The value of \(a\) cannot be zero, as this would make the equation linear, not quadratic. Quadratic equations often appear in problems involving projectile motion, area calculations, or optimization tasks. Solving quadratic equations helps us find the possible values of \(x\) that satisfy the equation. The quadratic formula is one of the primary methods used to solve these equations and handle complex problems.

Discriminant

The discriminant is a crucial component of the quadratic formula that helps in determining the nature of the roots of a quadratic equation. It is given by the expression:
\[ D = b^2 - 4ac \]
Depending on the value of the discriminant, we can learn about the roots without even solving for \(x\):

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, meaning the root is repeated (also known as a double root).
• If \(D < 0\), the equation has no real roots, but instead two complex roots.

In our example, after evaluating \(D\), we found it to be 3600, which is greater than zero, indicating two distinct real roots for the given equation.

Polynomial Coefficients

Polynomial coefficients are the numbers in front of the variables in a polynomial equation. These numbers can change, affecting the graph and solutions of the equation. In the context of quadratic equations:

• \(a\) is the coefficient of \(x^2\), which affects the parabola's width and direction (up or down).
• \(b\) is the coefficient of \(x\), influencing the vertex's symmetry and position.
• \(c\) is the constant term, determining the parabola's intersection with the y-axis.

In the equation \(32y^2 - 84y + 27 = 0\), the coefficients are \(a = 32\), \(b = -84\), and \(c = 27\). These numbers are critical because they are used in the quadratic formula to find the value of \(y\). Changing any of these coefficients will alter the solutions and shape of the equation's graph.