Problem 65 - Algebra for College Students

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

The value of \(k\) is either 4 or -4.

Step 1: Understand the Problem

To find the value of \(k\) such that the quadratic equation \(4x^2 - kx + 1 = 0\) has two equal real solutions, we need to explore the condition for a quadratic to have equal roots. This occurs when the discriminant is zero.

Step 2: Recall the Discriminant Formula

For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is given by \(\Delta = b^2 - 4ac\). The roots are equal when \(\Delta = 0\).

Step 3: Substitute into the Discriminant Formula

In our equation, \(a = 4\), \(b = -k\), and \(c = 1\). Substitute these into the discriminant formula: \[(-k)^2 - 4(4)(1) = 0\].

Step 4: Simplify the Equation

Simplify the discriminant equation: \[k^2 - 16 = 0\].

Step 5: Solve for \(k\)

Rearrange the equation to find \(k\): \[k^2 = 16\]. Take the square root of both sides: \[k = \pm 4\].

Step 6: Conclusion

The equation \(4x^2 - kx + 1 = 0\) has two equal real solutions when \(k = 4\) or \(k = -4\).

Key Concepts

The Discriminant

When solving quadratic equations, the discriminant provides valuable information about the nature of the roots. Its formula is \[ \Delta = b^2 - 4ac \]where

• \(a\), \(b\), and \(c\) represent the coefficients in the quadratic equation \(ax^2 + bx + c = 0\).
• \(b^2\) is the square of the coefficient of \(x\), which impacts how the parabola opens and its vertex.
• \(4ac\) involves both the coefficient of \(x^2\) and the constant term, balancing the quadratic equation.

The discriminant tells us the nature of the roots:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there are two equal real roots.
• If \(\Delta < 0\), the roots are complex and not real.

Therefore, analyzing the discriminant is crucial in determining the character and number of solutions a quadratic equation may have.

Equal Roots

Equal roots in a quadratic equation occur when the discriminant is zero. This is because having the discriminant equal to zero (\(\Delta = 0\)) eliminates the possibility of distinct or complex roots. In mathematical terms, this situation means \[b^2 - 4ac = 0\]. In this condition:

• The quadratic equation has what is called a 'perfect square trinomial'.
• The graph of the quadratic touches the x-axis at exactly one point, indicating the root's multiplicity of two.

To illustrate, consider a specific equation like \(4x^2 - kx + 1 = 0\):

• By substituting the given coefficients into \(\Delta = b^2 - 4ac\), you find:
• \((-k)^2 - 4(4)(1) = 0\) simplifies to \(k^2 - 16 = 0\).

Solving this equation tells us that \(k = 4\) or \(k = -4\), giving us the values where the quadratic will have equal roots. It's eye-opening how such a balance of coefficients leads to just one "double root."

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is derived directly from the process of completing the square and is applicable to any quadratic equation.The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula computes the roots \(x\) of any quadratic equation \(ax^2 + bx + c = 0\), regardless of the specific values of \(a\), \(b\), or \(c\). Here's how each part functions in finding roots:

• The term \(-b\) adjusts the form to find the middle of the quadratic expression.
• The square root component \(\sqrt{b^2 - 4ac}\) accounts for the separation between roots, linking back to the discriminant's role.
• The division by \(2a\) scales the result to correspond correctly with the quadratic's leading coefficient.

Notably, when \(b^2 - 4ac = 0\), the formula simplifies to a single root, confirming that the equation has equal roots.This special case illustrates the elegance and generality of the quadratic formula, making it essential for solving quadratic equations efficiently and accurately.