Problem 77 - Algebra 1: Concepts and Skills

Question

Tell whether the equation has two solutions, one solution, or no real solution. $$ x^{2}-5 x+6=0 $$

Step-by-Step Solution

Answer

The given equation has two real solutions.

Step 1: Identify the coefficients

From the given equation, identify the coefficients \(a\), \(b\), and \(c\). In this case, \(a=1\), \(b=-5\), and \(c=6\). These will be used to calculate the discriminant.

Step 2: Calculate the discriminant

Substitute the coefficients into the discriminant formula \(b^2 - 4ac\). Thus, the discriminant is \((-5)^2 - 4*1*6 = 25 - 24 = 1\)

Step 3: Determine the number of solutions

Since the discriminant is greater than 0, it can be concluded that the equation has two real solutions.

Key Concepts

The Discriminant in Quadratic Equations

Understanding the discriminant is essential when solving quadratic equations. It is the part of the quadratic formula under the square root, given by the expression \( b^2 - 4ac \). The discriminant can tell us a lot about the nature of the solutions without actually solving the equation.

For the quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated as follows:
\[ D = b^2 - 4ac \]
There are three scenarios based on the value of the discriminant:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution, sometimes referred to as a repeated or double root.
• If \( D < 0 \), the equation has no real solutions but two complex solutions.

For our example, \( x^2 - 5x + 6 = 0 \), the coefficients \( a = 1 \), \( b = -5 \), and \( c = 6 \), lead to a discriminant of \( D = 1 \), indicating two real solutions.

Solving with the Quadratic Formula

The quadratic formula is the solution to the quadratic equation \( ax^2 + bx + c = 0 \). This powerful tool provides the roots of any quadratic equation, as long as \( a \), \( b \), and \( c \) are known, and \( a \) is not zero.

The formula is:
\[ x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}} \]
The \( \pm \) symbol indicates that there are two solutions that come from taking the positive and negative square root of the discriminant. Using our previous example with the discriminant of \( 1 \), apply the quadratic formula:
\[ x = \frac{{-(-5) \pm \sqrt{1}}}{{2(1)}} \]
Which simplifies to:
\[ x = \frac{{5 \pm 1}}{2} \]
This yields the two solutions \( x = 3 \) and \( x = 2 \). It's a vital method for solving quadratics, and understanding it allows students to tackle any quadratic equation with confidence.

Number of Solutions in Quadratic Equations

The number of solutions to a quadratic equation can be determined using the discriminant, which gives us a quick answer to how many solutions we can expect.

The equation \( ax^2 + bx + c = 0 \) may yield:

• Two distinct real solutions when the discriminant \( D > 0 \).
• Exactly one real solution (a repeated root) when \( D = 0 \).
• No real solutions, but two complex solutions when \( D < 0 \).

In our exercise, the discriminant of \( 1 \) implies there are two distinct real solutions. Knowing this, students can proceed with confidence, applying the quadratic formula to find these solutions. The quadratic formula will always render the exact solutions whereas the discriminant provides a crucial, preliminary understanding of the solution set.