Problem 61 - Intermediate Algebra

Question

Your friend states that the equation \(-2 x^{2}+4 x-1=0\) must be changed to \(2 x^{2}-4 x+1=0\) (by multiplying both sides by \(-1\) ) before the quadratic formula can be applied. Is she right about this? If not, how would you convince her she is wrong?

Step-by-Step Solution

Answer

No, multiplying by -1 is unnecessary; the quadratic formula can be applied directly.

Step 1: Understand the Quadratic Formula

The quadratic formula is used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 2: Identify the Quadratic Equation

Examine the given equation \(-2x^2 + 4x - 1 = 0\). This is a quadratic equation where \(a = -2\), \(b = 4\), and \(c = -1\).

Step 3: Apply the Quadratic Formula

Using the coefficients \(a = -2\), \(b = 4\), and \(c = -1\), substitute them into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{4^2 - 4(-2)(-1)}}{2(-2)} \]Simplify the expression under the square root and the denominator.

Step 4: Simplify the Discriminant and Compute

Calculate the discriminant: \[ b^2 - 4ac = 4^2 - 4(-2)(-1) = 16 - 8 = 8 \]Thus, the quadratic formula becomes:\[ x = \frac{-4 \pm \sqrt{8}}{-4} \]Simplify further as required.

Step 5: Determine Conclusion

Verify if the original form needed alteration. Since the quadratic formula can be applied directly to the original equation, multiplying the equation by \(-1\) was unnecessary.

Key Concepts

Quadratic Equation

A quadratic equation is an equation in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This is the general form used to identify the coefficients necessary for applying mathematical methods to find its solutions.

These equations are called "quadratic" because they involve a squared term \(x^2\). Quadratic equations are crucial in algebra, and they represent parabolas when graphed on a coordinate plane.

• "\(a\)" is the coefficient of \(x^2\) and dictates the parabola's direction (upward if positive, downward if negative).
• "\(b\)" is the coefficient of \(x\) and affects the parabola's symmetry.
• "\(c\)" is the constant term representing the parabola's y-intercept.

In the exercise example, the quadratic equation is \(-2x^2 + 4x - 1 = 0\). It is important to understand this form to apply the quadratic formula properly without altering the equation unnecessarily.

Discriminant

The discriminant is a part of the quadratic formula represented by \(b^2 - 4ac\). It plays a crucial role in determining the nature and number of solutions of a quadratic equation.

The value of the discriminant provides insights:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (or repeated root).
• If \(b^2 - 4ac < 0\), no real roots exist, but two complex roots do.

For the quadratic equation in the exercise, we calculated the discriminant as follows:\[b^2 - 4ac = 16 - 8 = 8\]
This positive result (8) tells us that the equation has two distinct real roots.

Solving Quadratic Equations

Solving quadratic equations often involves the quadratic formula, which ensures any quadratic equation can find its roots. The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula directly uses the coefficients from the quadratic equation: \(a\), \(b\), and \(c\).

To solve the quadratic equation given in the exercise, we used these coefficients:

• \(a = -2\)
• \(b = 4\)
• \(c = -1\)

Even though it might seem tempting to rearrange or multiply the equation as suggested in the original problem, the formula applies seamlessly. Implementing it step-by-step, you insert the values into the formula:
\[x = \frac{-4 \pm \sqrt{8}}{-4}\]
Simplifying further delivers the solutions without needing to alter the equation beforehand. Thus, the quadratic formula proves convenient and efficient in solving these types of equations.