Problem 66 - Algebra 1

Question

Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. $$-2 x^{2}+6 x+1=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation -2x^2+6x+1=0 by using the quadratic formula are \(x_1=\frac{3 - \sqrt{11}}{2}\) and \(x_2=\frac{3 + \sqrt{11}}{2}\). The quadratic formula was chosen due to the discriminant not being a perfect square.

Step 1: Identify Coefficients

Identify the coefficients of the quadratic equation -2x^2+6x+1=0. Here, a=-2, b=6, and c=1.

Step 2: Calculate Discriminant

Calculate the discriminant using the formula b^2 - 4ac. Substituting the coefficients, we get (6)^2 - 4*(-2)*1=36 + 8=44.

Step 3: Choose a Method Based on Discriminant

44 is not a perfect square. Therefore the square root method would not provide a simple solution - the quadratic formula must be used instead.

Step 4: Solve Using Quadratic Formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the coefficients into the formula to find the solutions for x. \[x = \frac{-6 \pm \sqrt{44}}{-4}\]. This simplifies to \[x_1=\frac{3 - \sqrt{11}}{2}\] and \[x_2=\frac{3 + \sqrt{11}}{2}\]

Step 5: Check the Solutions

Substitute the solutions back into the original equation to check if they are correct.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra for solving quadratic equations. A quadratic equation is typically written in the form \( ax^2 + bx + c = 0 \). The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides a way to find the roots or solutions of any quadratic equation, whether simple or complex. It is derived from the process of completing the square, and it reliably gives the solutions for \( x \) by considering the coefficients \( a \), \( b \), and \( c \).

• It's essential when the quadratic equation does not factor easily.
• The formula takes into account the discriminant, which can help identify the nature of the roots.

Using the quadratic formula ensures that you can solve for both real and complex roots by substituting the specific values from the quadratic formula's coefficients.

Discriminant

The discriminant is a key feature within the quadratic formula. It is given by the expression \( b^2 - 4ac \). The value of the discriminant provides critical information about the nature of the roots of the equation.

• If the discriminant is positive, the equation has two real and distinct roots.
• If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the equation has two complex (non-real) roots.

For the equation \(-2x^2 + 6x + 1 = 0\), the discriminant is calculated as \( 36 + 8 = 44 \), which is positive and indicates two real and distinct roots. Understanding the discriminant's role helps in choosing the right method for solving quadratic equations.

Roots of Equations

Roots of an equation refer to the values of \( x \) that make the equation equal to zero. For quadratic equations, these roots are the solutions that satisfy \( ax^2 + bx + c = 0 \). The quadratic formula provides these roots: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In our problem, substituting the values gives:

• \( x_1 = \frac{3 - \sqrt{11}}{2} \)
• \( x_2 = \frac{3 + \sqrt{11}}{2} \)

This means both \( x_1 \) and \( x_2 \) are roots of the given quadratic equation. Each solution satisfies the equation, proving them as valid solutions. Checking these roots by substitution back into the original equation can confirm their correctness.

Coefficient Identification

Identifying coefficients is a fundamental first step in solving any quadratic equation. For an equation of the form \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) must be pinpointed correctly. In the problem \(-2x^2 + 6x + 1 = 0\):

• \( a \) is \(-2\)
• \( b \) is \(6\)
• \( c \) is \(1\)

These coefficients are crucial because they are directly substituted into the quadratic formula and the discriminant formula. Misidentifying these numbers can lead to incorrect solutions. They guide the direction for solving the equation, whether applying the quadratic formula or another method, such as factoring or completing the square.