Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(4 x=4 x^{2}-3\)

Step-by-Step Solution

Answer

The roots of the equation \(4x^2 - 4x - 3 = 0\) are \(x = 1\) and \(x = -0.75\)

Step 1: Rearrange the equation

The given equation is \(4 x = 4x^{2} - 3\). The first step is to rearrange this equation into the standard form of quadratic equation \( ax^2 + bx + c = 0 \). To do this, subtract \(4x\) from both sides to give \(4x^2 - 4x - 3 = 0\). The coefficients \(a\), \(b\), and \(c\) are 4, -4, and -3 respectively.

Step 2: Use Quadratic Formula

The quadratic formula is \(x=(-b±\sqrt{d})/(2a)\), where \(d = b^{2}-4ac\), the discriminant. The coefficients from the given quadratic equation are substituted into the quadratic formula to calculate the roots of the equation.

Step 3: Calculate the discriminant

Firstly, the discriminant \(d = b^{2}-4ac\) needs to be calculated. Here, \(b = -4\), \(a = 4\), and \(c = -3\). This gives \(d = ((-4)^{2}) - 4*4*(-3) = 16 + 48 = 64\)

Step 4: Substituting the discriminant

Substitute the discriminant \(d = 64\), \(a = 4\), and \(b = -4\) into the quadratic formula. The roots of the equation are given by \(x =(-b± \sqrt{d})/(2a) = ([4±\sqrt{64}]/[2*4]) , which simplifies to \(x = 1\) or \(x = -0.75\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is particularly useful when factoring is difficult or impossible. The formula is expressed as:

• \( x = \frac{-b \pm \sqrt{d}}{2a} \)

Here, \( d \) is known as the discriminant, which determines the nature and number of solutions of the quadratic equation. By substituting the coefficients \( a \), \( b \), and \( c \) of your quadratic equation into the formula, you can find the solutions, or "roots," of the equation. Quadratic equations may have two, one, or no real solutions, depending on the discriminant's value. This method is systematic and always works, making it a reliable way to solve quadratics.

Discriminant

The discriminant is an essential component of the quadratic formula, denoted by \( d \), and calculated as \( d = b^2 - 4ac \). It plays a crucial role in determining:

• The number of solutions the quadratic equation has.
• The type of solutions (real or complex).

To understand this better:

• If the discriminant \( d > 0 \), the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two points.
• If \( d = 0 \), there is exactly one real root, indicating the parabola touches the x-axis at a single point (a repeated or "double" root).
• If \( d < 0 \), there are no real roots; instead, there are two complex roots, meaning the parabola does not intersect the x-axis.

For instance, in the example equation \( 4x^2 - 4x - 3 = 0 \), the calculated discriminant was \( 64 \), which is positive, confirming two distinct real roots.

Roots of a Quadratic Equation

The "roots" of a quadratic equation are the solutions that satisfy the equation, essentially the values of \( x \) that make the equation true (i.e., equal to zero). These roots can be real or complex, depending on the value of the discriminant. Using the quadratic formula, once the discriminant is evaluated, substituting it back gives us the roots:

• For \( 4x^2 - 4x - 3 = 0 \), the discriminant was \( 64 \).
• Substituting into the formula, we have \( x = \frac{-(-4) \pm \sqrt{64}}{2 \times 4} \).
• This simplifies to \( x = \frac{4 \pm 8}{8} \).
• The solutions are \( x = 1 \) and \( x = -0.75 \).

Roots are important because they provide crucial information about the behavior and properties of quadratic functions, such as identifying x-intercepts of the graph of the quadratic function.