Problem 49 - Algebra A Combined Function

Question

Solve each equation. $$ 30 x^{2}-11 x=30 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{6}{5}\) and \(x = -\frac{5}{6}\).

Step 1: Write the Equation in Standard Form

The given equation is \(30x^2 - 11x = 30\). To solve this, let's first write it in the form \(ax^2 + bx + c = 0\) by moving all terms to one side. This yields: \[ 30x^2 - 11x - 30 = 0 \]

Step 2: Identify a, b, and c

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 30\), \(b = -11\), \(c = -30\).

Step 3: Use the Quadratic Formula

The quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) solves for \(x\). Substitute \(a = 30\), \(b = -11\), and \(c = -30\) into the formula: \[ x = \frac{{11 \pm \sqrt{{(-11)^2 - 4 \times 30 \times (-30)}}}}{2 \times 30} \]

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\): \((-11)^2 - 4 \times 30 \times (-30) = 121 + 3600 = 3721\)

Step 5: Simplify the Solutions

With the discriminant \(3721\), which is a perfect square, continue solving the quadratic formula: \[ x = \frac{{11 \pm \sqrt{3721}}}{60} \] Since \(\sqrt{3721} = 61\), substitute back: \[ x = \frac{11 + 61}{60} \quad \text{or} \quad x = \frac{11 - 61}{60} \]

Step 6: Calculate the Two Solutions

Solve for \(x\): First solution: \[ x = \frac{72}{60} = \frac{6}{5} \] Second solution: \[ x = \frac{-50}{60} = -\frac{5}{6} \]

Step 7: Final Step: Verify the Solutions

Verify by substituting back into the original equation: For \(x = \frac{6}{5}\): \[ 30 \left(\frac{6}{5}\right)^2 - 11 \left(\frac{6}{5}\right) = 30 \] For \(x = -\frac{5}{6}\): \[ 30 \left(-\frac{5}{6}\right)^2 - 11 \left(-\frac{5}{6}\right) = 30 \] Both verify the solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. This formula helps you find the solutions to quadratic equations in the form of \(ax^2 + bx + c = 0\). The formula itself is expressed as:

• \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)

To use this formula, identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. By substituting these values into the formula, you can compute the possible values for \(x\). These solutions are found by evaluating the expression under the square root, which is critical for determining the nature of the roots. This method of using the quadratic formula provides an exact solution when the equation is not easily factorable.

Discriminant

The discriminant is a key part of the quadratic formula and is located under the square root symbol, \(\sqrt{b^2 - 4ac}\). The value of the discriminant, \(b^2 - 4ac\), tells us important information about the solutions to a quadratic equation:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, the solutions are complex numbers.

In our example, the discriminant is calculated as \(((-11)^2 - 4 \times 30 \times (-30))\), which equals 3721—a positive number. This indicates that the quadratic equation has two distinct real solutions. Additionally, since 3721 is a perfect square (as \(61^2 = 3721\)), the solutions simplify easily.

Standard Form

To solve a quadratic equation using the quadratic formula or other methods, it must first be written in standard form. The standard form of a quadratic equation is:

• \(ax^2 + bx + c = 0\)

This allows you to clearly identify the coefficients \(a\), \(b\), and \(c\), which are essential for applying the quadratic formula. For our specific exercise, the original equation \(30x^2 - 11x = 30\) was rearranged to \(30x^2 - 11x - 30 = 0\). This transformation places the equation into the standard form by moving all terms to one side, allowing us to proceed with solving the equation accurately.

Solution Verification

After finding solutions to a quadratic equation, it is crucial to verify them. This verification checks whether the solutions satisfy the original equation. Let's break down the importance of this step:

• Substitute each solution back into the original equation.
• Solve the equation to confirm that both sides are equal.

In our example, the solutions found were \(x = \frac{6}{5}\) and \(x = -\frac{5}{6}\). We verify these solutions by substituting them back into the original equation \(30 x^{2}-11 x=30\):

• For \(x = \frac{6}{5}\), substitution into the original equation yields a true statement.
• Similarly, for \(x = -\frac{5}{6}\), substitution confirms the equation holds true.

This verification confirms that both solutions are correct and satisfy the original quadratic equation, ensuring accuracy in our solution process.