Problem 75 - Intermediate Algebra

Question

Use the discriminant to help solve each problem. Determine \(k\) so that \(4 x^{2}-k x+1=0\) has two equal real solutions.

Step-by-Step Solution

Answer

\(k = 4\) or \(k = -4\)

Step 1: Identify the coefficients

The given quadratic equation is \(4x^2 - kx + 1 = 0\). Let's write down the coefficients: \(a = 4\), \(b = -k\), and \(c = 1\).

Step 2: Recall the condition for equal real solutions

For a quadratic equation to have two equal real solutions, the discriminant \(D\) must be equal to zero. The discriminant for a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\).

Step 3: Set up the equation for the discriminant

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(D = (-k)^2 - 4 \cdot 4 \cdot 1\). Simplifying this gives \(D = k^2 - 16\).

Step 4: Solve for \(k\) such that the discriminant is zero

Set the discriminant equal to zero to find \(k\). Doing this yields the equation: \(k^2 - 16 = 0\). Solve for \(k\) by adding 16 on both sides, resulting in \(k^2 = 16\).

Step 5: Solve for \(k\)

Find \(k\) by taking the square root of both sides of \(k^2 = 16\). This gives two solutions: \(k = 4\) and \(k = -4\).

Key Concepts

Discriminant

The discriminant is a component in the quadratic equation that provides insight into the nature of the roots of the equation. Given the general form of a quadratic equation, \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). This value helps determine whether the roots of the quadratic equation are real or complex.

The discriminant can tell you:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, also known as a double root.
• If \(D < 0\), there are no real solutions; the roots are complex numbers.

For example, the discriminant \(D\) of the equation \(4x^2 - kx + 1 = 0\) is \(k^2 - 16\). By setting \(D = 0\), you can solve for \(k\) to ensure the equation has one real solution, indicating a double root.

Real Solutions

Real solutions refer to the values of \(x\) that satisfy the quadratic equation and are real numbers. By using the discriminant, you can predict whether a quadratic equation has real solutions and how many.

Remember:
- When \(D > 0\), the quadratic equation has two distinct real solutions. This occurs when the graph of the quadratic function intersects the \(x\)-axis at two different points.- When \(D = 0\), the equation has one real solution, or a repeated solution. This means the graph touches the \(x\)-axis at exactly one point, known as the vertex.- When \(D < 0\), there are no real solutions; instead, the solutions are complex and the graph does not intersect the \(x\)-axis.In our exercise, setting \(D = 0\) implies that we are looking for the condition in which the quadratic equation \(4x^2 - kx + 1 = 0\) has one real solution by making \(k = 4\) or \(k = -4\). This will produce a double root.

Coefficients

Coefficients in a quadratic equation are the numbers in front of the variables. In the equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are the coefficients. These values are crucial because they affect the shape of the graph of the equation and the nature of its solutions.

- \(a\) is the coefficient of \(x^2\), and it determines the direction of the parabola (whether it opens upwards or downwards).- \(b\) is the coefficient of \(x\), which affects the symmetry and position of the parabola.- \(c\) is the constant term, and it represents the point where the graph intersects the \(y\)-axis.In the given problem, the quadratic is \(4x^2 - kx + 1 = 0\). Here, \(a = 4\), \(b = -k\), and \(c = 1\). Identifying these coefficients correctly is essential for applying the discriminant formula and finding solutions for \(k\) to achieve specific types of real solutions.