Problem 31 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ x^{2}-7 x+10=0 $$

Step-by-Step Solution

Answer

The real solutions are \( x = 5 \) and \( x = 2 \).

Step 1: Identify the Equation Type

The equation given is a quadratic equation of the form \( ax^2 + bx + c = 0 \). In our case, \( a = 1 \), \( b = -7 \), and \( c = 10 \).

Step 2: Use the Quadratic Formula

To solve the quadratic equation \( x^2 - 7x + 10 = 0 \), we can apply the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is calculated as \( b^2 - 4ac \). Substitute the values: \( \Delta = (-7)^2 - 4 \times 1 \times 10 = 49 - 40 = 9 \).

Step 4: Compute the Square Root of the Discriminant

Since the discriminant \( \Delta = 9 \), compute its square root: \( \sqrt{9} = 3 \).

Step 5: Substitute into the Quadratic Formula

Now that we have \( \Delta \), substitute it into the formula: \[x = \frac{-(-7) \pm 3}{2 \times 1} = \frac{7 \pm 3}{2}.\]

Step 6: Solve for the Two Possible Solutions

The plus-minus operation gives two solutions:1. \( x_1 = \frac{7 + 3}{2} = \frac{10}{2} = 5 \).2. \( x_2 = \frac{7 - 3}{2} = \frac{4}{2} = 2 \).

Key Concepts

Quadratic Formula

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula is a tool that allows us to find the solutions for these types of equations. It is given by the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula provides a way to directly calculate the roots of the equation without necessarily factoring or graphing.
To use it effectively:

• First, identify the coefficients \( a \), \( b \), and \( c \) in the equation.
• Plug these values into the formula.
• Solve for \( x \) using basic arithmetic operations to find the solutions.

This method is especially helpful when the equation cannot be easily factored.

Discriminant

The discriminant is a key part of the quadratic formula. It is found under the square root in the formula: \( b^2 - 4ac \). This value helps determine the nature of the roots of the quadratic equation.
When calculating the discriminant:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution, which means both roots are the same.
• If it is negative, there are no real solutions, only complex solutions.

For our equation, \( \Delta = 9 \), indicating that there are two real solutions.

Real Solutions

Real solutions are the roots of the quadratic equation that can be plotted on a real-number line. They are the solutions that don't involve imaginary numbers. In our example, because the discriminant is positive (\( \Delta = 9 \)), we have two distinct real solutions: \( x_1 = 5 \) and \( x_2 = 2 \).
These solutions indicate the points where the quadratic polynomial intersects the x-axis. A positive discriminant ensures real roots, making the solutions observable in real-world contexts, such as projectile paths or economics models.

Step by Step Solutions

Step by step solutions provide clarity on why and how each part of the problem is solved. This approach breaks down complex processes into more manageable parts.
For the quadratic equation \( x^2 - 7x + 10 = 0 \):

• We identified it as a quadratic equation with \( a = 1 \), \( b = -7 \), and \( c = 10 \).
• Calculated the discriminant \( b^2 - 4ac \), which was 9, a positive number.
• Used the quadratic formula, substituting \( \, b = -7 \, \) and \( \, c = 10 \, \) to find the roots.
• Determined the solutions \( x_1 = 5 \) and \( x_2 = 2 \) by computing \( \frac{7 \pm 3}{2} \).

This detailed breakdown ensures each decision is understood fully, allowing for similar equations to be solved more confidently in the future.