Problem 58 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 2 x^{2}-3 x=14 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3.5\) and \(x = -2\).

Step 1: Move All Terms to One Side

Start by moving all terms to one side of the equation to form a quadratic equation in the standard form, which is \(ax^2 + bx + c = 0\). Move the constant from the right side of the equation to the left by subtracting 14: \[ 2x^2 - 3x - 14 = 0 \]

Step 2: Identify Coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(2x^2 - 3x - 14 = 0\). Here, \(a = 2\), \(b = -3\), and \(c = -14\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula to find the solutions for \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Plug in the values of \(a\), \(b\), and \(c\): \[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-14)}}{2 \cdot 2} \]

Step 4: Calculate the Discriminant

Calculate the value inside the square root, known as the discriminant, \(b^2 - 4ac\): \[ (-3)^2 - 4 \times 2 \times (-14) = 9 + 112 = 121 \]

Step 5: Calculate the Square Root of the Discriminant

Find the square root of the discriminant: \[ \sqrt{121} = 11 \]

Step 6: Find the Values of x

Substitute the value of the square root back into the quadratic formula to solve for \(x\): First solution: \[ x = \frac{3 + 11}{4} = \frac{14}{4} = 3.5 \]Second solution: \[ x = \frac{3 - 11}{4} = \frac{-8}{4} = -2 \]

Step 7: Verify the Solutions

Substitute both solutions back into the original equation to ensure they are correct:- For \(x = 3.5\):\(2(3.5)^2 - 3(3.5) = 24.5 - 10.5 = 14\), which is correct.- For \(x = -2\):\(2(-2)^2 - 3(-2) = 8 + 6 = 14\), which is correct. Therefore, both solutions are verified.

Key Concepts

Quadratic Formula

The quadratic formula is one of the most vital tools for solving quadratic equations, which are equations characterized by the form \( ax^2 + bx + c = 0 \). The formula provides a straightforward method to find the solutions (or roots) of these types of equations. It is expressed as follows:

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

This means that you take the opposite of the coefficient \( b \), add and subtract the square root of the discriminant \( b^2 - 4ac \), and divide by twice the coefficient \( a \). The use of "plus-minus" (±) sign indicates that there can be two potential solutions, corresponding to the "plus" and "minus".
In our problem, after rewriting the equation in standard form, we identify \( a = 2 \), \( b = -3 \), and \( c = -14 \), and put these into the formula, allowing us to compute the specific solutions quickly.

Discriminant

The discriminant is the component of the quadratic formula under the square root: \( b^2 - 4ac \). This part is crucial because it reveals the nature of the roots of the quadratic equation:

• If the discriminant is positive, there are two real and distinct solutions.
• If it’s zero, there is exactly one real solution (the roots are repeated).
• If it’s negative, there are no real solutions, though there are complex ones.

For our specific equation, the discriminant calculation is:

• \((-3)^2 - 4 \, \cdot \, 2 \, \cdot \, (-14) = 9 + 112 = 121\)

A positive discriminant of 121 means our equation has two distinct real solutions. Solving \( \sqrt{121} \) yields 11, further allowing us to complete the quadratic formula process.

Solution Verification

Once you calculate potential solutions to the equation using the quadratic formula, it's crucial to verify them. Verification ensures that the solutions are correct by substituting back into the original equation. If both sides of the equation are equal after substitution, the solutions are validated.
For our problem, the solutions found are \( x = 3.5 \) and \( x = -2 \). To verify:

• Substitute \( x = 3.5 \) into the equation: \( 2(3.5)^2 - 3(3.5) = 24.5 - 10.5 = 14 \)
• Substitute \( x = -2 \) into the equation: \( 2(-2)^2 - 3(-2) = 8 + 6 = 14 \)

Both substitutions confirm the equation equals 14, the same as our original equation, proving both solutions are correct. Always take this step to ensure no errors occurred during calculation.