Problem 70 - Algebra for College Students

Question

Solve each of the following equations for \(x\). $$10 x^{2}-31 a x-14 a^{2}=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{7a}{2} \) and \( x = -\frac{2a}{5} \).

Step 1: Identify the type of equation

The given equation is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 10 \), \( b = -31a \), and \( c = -14a^2 \).

Step 2: Use the quadratic formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) will be used to solve the equation. Substitute the values \( a = 10 \), \( b = -31a \), and \( c = -14a^2 \) into the formula.

Step 3: Calculate the discriminant

Calculate the discriminant \( abla = b^2 - 4ac = (-31a)^2 - 4 \times 10 \times (-14a^2) \). Therefore, \(abla = 961a^2 + 560a^2 = 1521a^2\).

Step 4: Solve for x using the quadratic formula

Since \( abla = 1521a^2 \) is a perfect square \( (39a)^2 \), use the quadratic formula: \( x = \frac{-(-31a) \pm 39a}{20} \). This results in \( x = \frac{31a \pm 39a}{20} \).

Step 5: Simplify the expression

Calculate the two possible solutions: \( x = \frac{70a}{20} = \frac{7a}{2} \) and \( x = \frac{-8a}{20} = -\frac{2a}{5} \).

Step 6: Conclude the solutions

The solutions for the equation \( 10x^2 - 31ax - 14a^2 = 0 \) are \( x = \frac{7a}{2} \) and \( x = -\frac{2a}{5} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula provides the solution for \( x \) by using the coefficients \( a \), \( b \), and \( c \) directly. It is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula calculates two potential solutions for \( x \) (hence the plus-minus symbol) and applies to any quadratic equation, whether its coefficients are real numbers, fractions, or expressions involving variables.
When using the formula, it is important to correctly identify and substitute the values of \( a \), \( b \), and \( c \). For instance, in the equation \( 10x^2 - 31ax - 14a^2 = 0 \), the values are \( a = 10 \), \( b = -31a \), and \( c = -14a^2 \). Ensuring accurate substitution is crucial for solving the equation correctly.

Discriminant Calculation

To determine the nature and number of solutions for a quadratic equation, calculating the discriminant \( \Delta \) is essential. The discriminant is the part under the square root in the quadratic formula:
\[\Delta = b^2 - 4ac\]
Here, the discriminant provides critical information:

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution (also called a repeated or double root).
• If \( \Delta < 0 \), there are two complex solutions that aren't real.

In the given equation \( 10x^2 - 31ax - 14a^2 = 0 \), substituting into the discriminant formula gives:
\[\Delta = (-31a)^2 - 4 \times 10 \times (-14a^2)\]
Calculating each part, we find:
\[(-31a)^2 = 961a^2 \quad \text{and} \quad 4 \times 10 \times (-14a^2) = -560a^2\]
Thus, \( \Delta = 961a^2 + 560a^2 = 1521a^2 \). Because this is a perfect square \( (39a)^2 \), the equation has two distinct real solutions.

Solving Quadratic Equations

Once you have calculated the discriminant and substituted the coefficients into the quadratic formula, the next step is solving for \( x \). Solving involves carrying out the operations prescribed by the quadratic formula:
For the equation \( 10x^2 - 31ax - 14a^2 = 0 \), we substitute into the formula:
\[x = \frac{-(-31a) \pm 39a}{20}\]
This simplifies to:
\[x = \frac{31a \pm 39a}{20}\]
This step yields two possible solutions by adding and subtracting the terms:

• First solution: \( x = \frac{31a + 39a}{20} = \frac{70a}{20} = \frac{7a}{2} \)
• Second solution: \( x = \frac{31a - 39a}{20} = \frac{-8a}{20} = -\frac{2a}{5} \)

These results show that solving quadratic equations involves precise calculation and simplification to arrive at the correct solutions. Knowing how to apply the quadratic formula step by step allows for solving any quadratic equation with confidence.