Problem 31 - Algebra 1

Question

Decide how many solutions the equation has. $$x^{2}-2 x+4=0$$

Step-by-Step Solution

Answer

The equation \(x^{2}-2 x+4=0\) has no real solutions.

Step 1 Identify coefficients a, b and c

In the quadratic equation \(ax^{2} + bx + c = 0\), the coefficient \(a\) is the number in front of \(x^{2}\), \(b\) in front of \(x\) and \(c\) is the constant. For the equation \(x^{2} - 2x + 4 = 0\), the coefficients are: \(a = 1\), \(b = -2\), and \(c = 4\).

Step 2 Calculate the Discriminant

Using the formula for discriminant \(D = b^{2}-4ac\) with \(a = 1\), \(b = -2\), and \(c = 4\), we substitute those values in and get \(D = (-2)^{2}-4*1*4\).

Step 3 Determine the Number of Solutions

From the previous step, we calculate \(D = 4 - 16 = -12\). Since \(D < 0\), there are no real solutions for the given equation.

Key Concepts

Discriminant

The discriminant is a vital part of solving quadratic equations. It tells us how many solutions, or roots, an equation has.
In the standard form of a quadratic equation, which is given by \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula:
\[D = b^2 - 4ac\]
By simply plugging the coefficients into this formula, you can determine the nature of the roots without actually solving the equation completely. Here's what the value of \(D\) indicates:

• **Positive Discriminant (\(D > 0\))**: There are two distinct real solutions.
• **Zero Discriminant (\(D = 0\))**: There is exactly one real solution, also known as a repeated or double root.
• **Negative Discriminant (\(D < 0\))**: There are no real solutions, meaning the solutions are complex numbers.

These outcomes help you understand if a quadratic equation can be factored into real solutions or if it needs a different approach.

Real Solutions

Real solutions to a quadratic equation occur when the solutions are real numbers, which is something most introductory problems focus on.
For an equation to have real solutions, its discriminant (\(D\)) must be greater than or equal to zero. This means there is no 'imaginary' number component.
When \(D > 0\), the quadratic equation yields two different real numbers for values of \(x\), which can be distinctly seen on a graph as the points where the parabola crosses the x-axis.
If \(D = 0\), it means the parabola just touches the x-axis at one point, which is the vertex; thus, there is one real, repeated solution.
On the other hand, if \(D < 0\), as seen in the equation \(x^2 - 2x + 4 = 0\), there are no points where the parabola crosses or touches the x-axis, resulting in no real solutions.

Coefficients

Coefficients in a quadratic equation provide the foundation for determining its solutions.
They are the numerical factors in front of the variables in the equation \(ax^2 + bx + c = 0\).

• **\(a\)** is the coefficient of \(x^2\). It determines the parabola's direction. If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards.
• **\(b\)** is the coefficient of \(x\). It affects the axis of symmetry of the parabola.
• **\(c\)** is the constant term. It represents the point where the parabola intersects the y-axis.

In our exercise, the coefficients were identified as \(a = 1\), \(b = -2\), and \(c = 4\). These values were used in the discriminant formula to determine that \(D = -12\), leading us to conclude there are no real solutions for the equation \(x^2 - 2x + 4 = 0\). The coefficients not only aid in computations but also give insight into the graph's behavior even before solving the equation.