Problem 51 - Algebra for College Students

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

She is wrong; the quadratic formula applies directly to the given equation.

Step 1: Write Down the Given Equation

The given quadratic equation is \[-2x^2 + 4x - 1 = 0\].

Step 2: Recall the Quadratic Formula

The quadratic formula is used to solve equations of the form \(ax^2 + bx + c = 0\) and is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 3: Identify the Coefficients for the Quadratic Formula

For the equation \(-2x^2 + 4x - 1 = 0\), assign the coefficients:- \(a = -2\) - \(b = 4\) - \(c = -1\).

Step 4: Check Applicability of Quadratic Formula

The quadratic formula applies to any quadratic equation of the form \(ax^2 + bx + c = 0\), regardless of whether the leading coefficient \(a\) is negative or positive. Thus, the equation \(-2x^2 + 4x - 1 = 0\) does not need to be multiplied by \(-1\).

Step 5: Conclusion

The friend is incorrect in saying that multiplication by \(-1\) is necessary. The quadratic formula can indeed be applied directly to \(-2x^2 + 4x - 1 = 0\) without any alterations.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve any quadratic equation. A quadratic equation is generally in the form of \( ax^2 + bx + c = 0 \). However, the key to understanding the quadratic formula lies in manipulating these equations to find their roots or solutions.
The formula itself is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This equation helps us find the values of \( x \) that make the original equation valid or true.

• The "\( b^2 - 4ac \)" part of the formula is called the discriminant. It determines the nature of the roots (real or complex).
• The "\( \pm \)" sign shows that there can be two possible solutions because it accommodates both addition and subtraction.
• Remember, the quadratic formula works regardless of whether the coefficient \( a \) is positive or negative.

Understanding this formula allows students to solve various quadratic equations, regardless of their complexity.

Coefficients Identification

Identifying coefficients is a crucial step in applying the quadratic formula. Coefficients \( a \), \( b \), and \( c \) are the numbers multiplying \( x^2 \), \( x \), and the constant term respectively in any quadratic equation of the form \( ax^2 + bx + c = 0 \).
For the specific exercise with the equation \(-2x^2 + 4x - 1 = 0\):

• \( a = -2 \)
• \( b = 4 \)
• \( c = -1 \)

These coefficients are plugged into the quadratic formula to determine the value of \( x \). It is important to identify these correctly as any mistake could lead to incorrect results in solving equations. Notice that the negative sign before \( a \) doesn't impede the use of the quadratic formula.

Solving Quadratic Equations

Now that we've got our coefficients lined up, the next task is solving the quadratic equation using the identified coefficients. Using our exercise example, once we've identified \( a = -2 \), \( b = 4 \), and \( c = -1 \), we substitute them into the quadratic formula.
Here's how the process unfolds:

• Plug \( a \), \( b \), and \( c \) into the discriminant part of the quadratic formula: \( b^2 - 4ac \).
• Calculate the discriminant to determine the nature of the roots. If it's positive, the roots are real and distinct.
• Substitute the coefficients into the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for the exact values of \( x \).

Trying it out with negative leading coefficients may seem intimidating at first, but remember, multiplication of an equation by a constant like -1 is not necessary unless specifically required by the problem in context. Using the quadratic formula works effectively with negative values, meaning that each equation you tackle brings you a step closer to mastering quadratic equations.