Problem 26 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$4 x^{2}-12 x=7$$

Step-by-Step Solution

Answer

The solutions are \(x = 3.5\) and \(x = -0.5\).

Step 1: Standard Form of a Quadratic Equation

Rewrite the given equation \(4x^2 - 12x = 7\) in the standard form of a quadratic equation, \(ax^2 + bx + c = 0\). Subtract 7 from both sides to get \(4x^2 - 12x - 7 = 0\).

Step 2: Identify Coefficients

Identify the coefficients in the quadratic equation \(4x^2 - 12x - 7 = 0\). Here, \(a = 4\), \(b = -12\), and \(c = -7\).

Step 3: Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 4\), \(b = -12\), and \(c = -7\) into this formula. This gives \(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-7)}}{2 \cdot 4}\).

Step 4: Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac\). For our equation, \((-12)^2 - 4 \cdot 4 \cdot (-7) = 144 + 112 = 256\).

Step 5: Solve the Equation

With the discriminant calculated as 256, substitute back into the quadratic formula to solve for \(x\). This results in \(x = \frac{12 \pm \sqrt{256}}{8}\).

Step 6: Simplify the Solutions

Calculate \(\sqrt{256} = 16\). Substitute this back into the equation: \(x = \frac{12 \pm 16}{8}\). This gives two solutions: \(x = \frac{28}{8} = 3.5\) and \(x = \frac{-4}{8} = -0.5\).

Key Concepts

Quadratic Equation

Quadratic equations are fundamental in algebra. They are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants known as the coefficients, and \(x\) represents the variable. The highest exponent in a quadratic equation is 2, which gives it the name "quadratic." It forms a parabola when graphed. Quadratic equations can have:

• two real roots
• one real root
• no real roots (when the solutions are complex numbers)

Understanding quadratic equations is essential for solving various mathematical problems. The quadratic formula is a powerful tool that helps find the roots of these equations.

Discriminant

In the realm of quadratic equations, the discriminant is a crucial concept. It is denoted by the expression \(b^2 - 4ac\). The value of the discriminant helps determine the nature and number of the roots of a quadratic equation. Here's how it works:

• If the discriminant is positive, the quadratic equation has two distinct real roots. For example, in our problem, the discriminant is 256, which is positive, implying two real solutions: 3.5 and -0.5.
• If the discriminant is zero, the quadratic equation has exactly one real root, meaning the parabola touches the x-axis at one point.
• If the discriminant is negative, there are no real roots; instead, the solutions are complex numbers.

Understanding the discriminant helps to quickly know how many and what kind of solutions to expect before diving into calculations.

Coefficients

The coefficients in a quadratic equation are vital parts that define its shape and position. In the equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are coefficients:

• \(a\) is the quadratic coefficient, determining the parabola's direction (upward if positive, downward if negative).
• \(b\) is the linear coefficient, which affects the position of the vertex along the x-axis.
• \(c\) is the constant term, which gives the y-intercept of the parabola when the equation is graphed.

Correctly identifying these coefficients is crucial in solving quadratic equations using the quadratic formula. For example, in the equation \(4x^2 - 12x - 7 = 0\), \(a = 4\), \(b = -12\), and \(c = -7\). These values are substituted into the quadratic formula to find the solutions.