Question

Challenge Problems $$2.96 x^{2}-33.2 x+4.05=0$$

Short answer

The roots of the equation \(2.96x^2 - 33.2x + 4.05 = 0\) are found using the quadratic formula and depend on the value of the discriminant, calculated as \(-33.2^2 - 4 \times 2.96 \times 4.05\).

Step-by-step solution

Identify the quadratic equation

The given exercise is a quadratic equation in the form of \(ax^2 + bx + c = 0\), where \(a = 2.96\), \(b = -33.2\), and \(c = 4.05\).

Apply the quadratic formula

To find the roots of the quadratic equation, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(x\) represents the roots of the equation.

Calculate the discriminant

Calculate the discriminant \(D = b^2 - 4ac\) to determine the nature of the roots. In this case, \(D = (-33.2)^2 - 4 \times 2.96 \times 4.05\).

Evaluate the roots

After finding the discriminant, evaluate the roots using the quadratic formula: \(x = \frac{33.2 \pm \sqrt{D}}{5.92}\).

Simplify the roots

Simplify the expression to find the exact values of \(x\). If the discriminant is positive, there will be two real roots. If it is zero, there will be one real root (a repeated root). If it is negative, there will be two complex roots.

Key concepts

Quadratic Formula

Understanding the quadratic formula is like having a master key for unlocking the solutions to a wide variety of quadratic equations. It represents a surefire method to find the roots of any quadratic equation, which is essentially a polynomial equation of the second degree typically expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).

The magic formula that allows us to find the roots is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This equation reveals the possible values of \(x\) that satisfy the original quadratic equation. Applying this formula involves a series of substitution steps:

• Insert the coefficients \(a\), \(b\), and \(c\) from the equation into the formula.
• Calculate the discriminant, which is the part under the square root sign \(\sqrt{b^2 - 4ac}\).
• Apply the plus or minus (\(\pm\)) sign to account for two possible values of \(x\).
• Divide the result by \(2a\) to find the final solutions for \(x\).

For students looking to grasp this concept better, it's essential to practice applying the quadratic formula to multiple problems to become familiar with the process and to recognize patterns in the types of solutions they encounter.

Discriminant

The discriminant in a quadratic equation is a beacon that shines light on the nature of the roots without actually solving for them. Represented by \(D\), it is found using the formula \(D = b^2 - 4ac\), which uses the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

The value of the discriminant gives us critical information:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is one real root, also known as a repeated or double root.
• If \(D < 0\), the roots are complex and involve imaginary numbers.

Understanding the discriminant helps students preemptively understand the results they will get before diving into the actual solution. It is a valuable tool for anticipating the character of the solutions and can serve as a quick check to ensure the solutions calculated from the quadratic formula are reasonable.

Roots of a Quadratic Equation

The roots of a quadratic equation are the x-values where the quadratic graph intersects the x-axis. To put it simply, these roots are the solutions to the equation. They can be real or complex numbers depending on the discriminant's value as it dictates the root type.

For \(ax^2 + bx + c = 0\), the roots can be calculated using the quadratic formula explained previously. If the roots are real, they represent the x-coordinates of the points at which the quadratic curve touches or crosses the x-axis. These points are important as they can indicate maximum or minimum values of the quadratic function depending on the direction of the parabola (upward or downward).

Exercise Improvement Advice

To further aid students in understanding the roots:

• Visualize the parabola graphically to see where it intersects the x-axis.
• Associate the discriminant's nature with the graph's intersection points.
• Perform additional practice with quadratic equations of varying coefficients to see how different root types manifest.
• Remember that complex roots come in conjugate pairs, which means if \(a + bi\) is a root, then \(a - bi\) is also a root, where \(i\) is the square root of negative one.

By understanding these concepts and utilizing the quadratic formula with the discriminant, students can tackle any quadratic equation confidently.