Problem 53 - Algebra A Combined Function

Question

Solve each equation. $$ 6 y^{2}-22 y-40=0 $$

Step-by-Step Solution

Answer

The solutions are \(y = 5\) and \(y = -\frac{4}{3}\).

Step 1: Identify the Type of Equation

The equation given is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 6\), \(b = -22\), and \(c = -40\).

Step 2: Use the Quadratic Formula

We use the quadratic formula, \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve for \(y\). Plugging in the values of a, b, and c gives us:\[y = \frac{-(-22) \pm \sqrt{(-22)^2 - 4(6)(-40)}}{2(6)}\]

Step 3: Calculate the Discriminant

Calculate \(b^2 - 4ac\) to find the discriminant:\((-22)^2 = 484\) and \(4 \times 6 \times 40 = 960\), so the discriminant is \(484 + 960 = 1444\).

Step 4: Evaluate the Square Root of the Discriminant

Compute the square root of the discriminant: \(\sqrt{1444} = 38\).

Step 5: Solve for 'y' Using the Quadratic Formula

Substitute the square root back into the quadratic formula:\[y = \frac{22 \pm 38}{12}\]This gives us two solutions:1. \(y = \frac{22 + 38}{12} = \frac{60}{12} = 5\)2. \(y = \frac{22 - 38}{12} = \frac{-16}{12} = -\frac{4}{3}\)

Step 6: Write the Final Solutions

The solutions to the equation \(6y^2 - 22y - 40 = 0\) are \(y = 5\) and \(y = -\frac{4}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is one of the most powerful tools for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the solutions for \( x \) directly, regardless of whether the quadratic can be factored easily or not. The quadratic formula is written as: \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are constants from the quadratic equation. This formula is beloved because it works with any quadratic equation, providing two potential solutions. These solutions correspond to the parabola's points at which it intersects the x-axis.

• Useful for any quadratic equation
• Provides a way to find exact roots
• Essential for equations that cannot be factored easily

You can always rely on this formula to find real or complex roots of quadratic equations as it considers all possible configurations including those with complex numbers.

Discriminant of Quadratic Equation

The discriminant is a special value in the quadratic formula that helps us understand the nature of the roots of a quadratic equation. It is the part inside the square root: \[b^2 - 4ac\] The discriminant tells us several key things about the equation:

• If the discriminant is positive, \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If the discriminant is zero, \(b^2 - 4ac = 0\), the equation has exactly one real root (roots are repeated).
• If the discriminant is negative, \(b^2 - 4ac < 0\), the equation has two complex and conjugate roots.

For the given problem, we calculated the discriminant to be 1444, which is positive, indicating that our quadratic equation has two distinct real roots. This helps in planning the next steps, as it confirms that actual real number calculations would eventually lead to our solution.

Square Root Calculation

The calculation of the square root is a critical part of solving quadratic equations using the quadratic formula. This arises when evaluating the discriminant, \( \sqrt{b^2 - 4ac} \). Calculating the square root correctly is crucial because it influences the determination of the roots' values. Whenever the discriminant is calculated, the next step is to find its square root. For our problem, the discriminant was \(1444\), and its square root calculates to \(38\). Correctly finding this value ensures that both roots determined via the quadratic formula are accurate.

• Plays a key role in evaluating quadratic roots
• Essential for precise calculation of roots' values
• Affects the accuracy of the overall solution

In the end, correct square root calculations ensure that you know exactly the nature and values of the solutions that the quadratic equation will yield.