Problem 126 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(4 d^{2}-7 d+2=0\)

Step-by-Step Solution

Answer

d = \(\frac{7 \pm \sqrt{17}}{8}\)

Step 1: Identify coefficients a, b, and c

In the quadratic equation of the form ax^2 + bx + c = 0the coefficients are the numbers a, b, and c. For the equation 4d^2 - 7d + 2 = 0,a = 4, b = -7, and c = 2.

Step 2: Apply the Quadratic Formula

The Quadratic Formula is given by \(d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Substitute the values of a, b, and c into the formula: \(d = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(2)}}{2(4)}\).

Step 3: Calculate the Discriminant

The discriminant \(b^2 - 4ac\)helps determine the nature of the roots. Calculate it as follows: \((-7)^2 - 4(4)(2) = 49 - 32 = 17.\)

Step 4: Solve for d

Substitute the discriminant back into the quadratic formula: \(d = \frac{7 \pm \sqrt{17}}{8}\). Thus, the solutions are \(d = \frac{7 + \sqrt{17}}{8}\) and \(d = \frac{7 - \sqrt{17}}{8}\).

Key Concepts

coefficients

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the coefficients are crucial components. They are the numbers that appear in front of the variables. The coefficient \(a\) is the number before \(x^2\), \(b\) is the number before \(x\), and \(c\) is the constant term.

In the equation given in the problem, \(4d^2 - 7d + 2 = 0\), we identify:

• a = 4 (coefficient of \(d^2\))
• b = -7 (coefficient of \(d\))
• c = 2 (constant term)

Recognizing these coefficients correctly is the first step in solving the quadratic equation using the quadratic formula.

discriminant

The discriminant is a key part of the quadratic formula, as it reveals the nature of the equation's roots. The discriminant is given by the expression \(b^2 - 4ac\).

For our example, we calculate the discriminant with the values of \(a\), \(b\), and \(c\) identified earlier:
\[(-7)^2 - 4(4)(2) = 49 - 32 = 17\]
The value of 17 indicates that the quadratic equation has two distinct real roots since the discriminant is positive.

Whenever you solve a quadratic equation, always start by calculating the discriminant because it tells you:

• If the roots are real or complex
• If the roots are distinct or repeated

roots of a quadratic equation

The roots of a quadratic equation are the values of the variable that satisfy the equation. They are found using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our example, we substitute \(a = 4\), \(b = -7\), and \(c = 2\) into the formula:
\[d = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(2)}}{2(4)} = \frac{7 \pm \sqrt{17}}{8}\]
These calculations provide us with two roots:

• \(d = \frac{7 + \sqrt{17}}{8}\)
• \(d = \frac{7 - \sqrt{17}}{8}\)

Knowing the roots helps us understand the points where the equation intersects the x-axis when graphed.

solving quadratic equations

Solving quadratic equations effectively involves multiple steps: identifying coefficients, calculating the discriminant, and using the quadratic formula to find the roots. Let’s go over these steps with our example equation:

• Start by identifying the coefficients \(a\), \(b\), and \(c\). For \(4d^2 - 7d + 2 = 0\), we have \(a = 4\), \(b = -7\), and \(c = 2\).
• Next, calculate the discriminant using \(b^2 - 4ac\). For our equation, it is \(17\).
• Lastly, substitute these values into the quadratic formula: \(d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to get the solutions: \(d = \frac{7 \pm \sqrt{17}}{8}\).

Following these steps systematically helps to accurately find the roots, ensuring you solve the equation correctly.