Problem 47 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$-2 x^{2}+4 x-3=0$$

Step-by-Step Solution

Answer

The equation has complex roots: \(1 - \frac{i\sqrt{2}}{2}\) and \(1 + \frac{i\sqrt{2}}{2}\).

Step 1: Identify the coefficients

First, identify the coefficients of the quadratic equation. The equation is \(-2x^2 + 4x - 3 = 0\). So, \(a = -2\), \(b = 4\), and \(c = -3\).

Step 2: Write the quadratic formula

The quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] We will use this formula to find the roots of the equation.

Step 3: Calculate the discriminant

The discriminant is the part of the formula under the square root; it's \(b^2 - 4ac\). Substitute \(a = -2\), \(b = 4\), and \(c = -3\) into the discriminant: \[ 4^2 - 4(-2)(-3) = 16 - 24 = -8 \]

Step 4: Analyze the discriminant

Since the discriminant is \(-8\), and it is less than zero, this indicates that the quadratic equation has no real roots. However, it will have two complex roots.

Step 5: Solve for Roots

Substitute the values \(a = -2\), \(b = 4\), \(c = -3\), and the discriminant into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{-8}}{-4} \] Simplify further: The square root of \(-8\) can be expressed as \( \sqrt{-8} = \sqrt{-1 \times 8} = i\sqrt{8} = 2i\sqrt{2} \). So, the expression becomes \[ x = \frac{-4 \pm 2i\sqrt{2}}{-4} \] Simplifying gives: \[ x_1 = \frac{-4 + 2i\sqrt{2}}{-4} = 1 - \frac{i\sqrt{2}}{2} \] and \[ x_2 = \frac{-4 - 2i\sqrt{2}}{-4} = 1 + \frac{i\sqrt{2}}{2} \] These are the two complex roots.

Step 6: Verify with sum and product of roots

For a quadratic equation \(ax^2 + bx + c = 0\), the sum of roots is \(\frac{-b}{a}\) and the product of roots is \(\frac{c}{a}\).- Sum of roots: \(1 - \frac{i\sqrt{2}}{2} + 1 + \frac{i\sqrt{2}}{2} = 2\)- The calculated sum is \(-\frac{b}{a} = \frac{-4}{-2} = 2\), which matches.- Product of roots: \((1 - \frac{i\sqrt{2}}{2})(1 + \frac{i\sqrt{2}}{2}) = 1 - (\frac{i\sqrt{2}}{2})^2 = 1 + \frac{2}{4} = 1.5\).- The calculated product is \(-\frac{3}{-2} = 1.5\), which matches. Verification confirms the roots satisfy the relationships for sum and product.

Key Concepts

Complex Roots

Complex roots arise when solving quadratic equations whose discriminant is less than zero. This happens because the discriminant, which is the part of the quadratic formula under the square root, becomes negative. In the quadratic equation \(-2x^2 + 4x - 3 = 0\), the discriminant is calculated as follows:\[ b^2 - 4ac = 4^2 - 4(-2)(-3) = 16 - 24 = -8 \] Since the discriminant is \(-8\), it is negative, indicating complex roots. Complex roots are expressed in terms of imaginary numbers, where \(i\) is the imaginary unit satisfying \(i^2 = -1\). Simplifying \(\sqrt{-8}\) reveals the imaginary part:

• \(\sqrt{-8} = \sqrt{-1 \times 8} = i\sqrt{8} = 2i\sqrt{2}\)

Thus, the roots obtained are:

• \(x_1 = 1 - \frac{i\sqrt{2}}{2}\)
• \(x_2 = 1 + \frac{i\sqrt{2}}{2}\)

These results demonstrate that the quadratic equation has no real solutions, only complex ones, due to the negative discriminant.

Discriminant

The discriminant is a key part of the quadratic formula used to determine the nature of the roots of a quadratic equation. It is calculated using the expression \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the equation. The discriminant can reveal crucial information about the roots:

• If it is greater than zero, the quadratic equation has two distinct real roots.
• If it is zero, the equation has two identical real roots, also known as a repeated root.
• If it is less than zero, as in the equation \(-2x^2 + 4x - 3 = 0\), it indicates the presence of complex roots.

In this example, the discriminant calculation yields \(-8\), a negative value, confirming the equation's complex roots. Knowing the discriminant helps quickly identify the type of roots to expect before further calculations.

Sum and Product of Roots

The sum and product of the roots provide valuable insights and a method for verifying the accuracy of the calculated roots for quadratic equations. For any quadratic equation \(ax^2 + bx + c = 0\), the sum and product of its roots \(x_1\) and \(x_2\) relate to the coefficients:

• The sum of the roots \(x_1 + x_2 = \frac{-b}{a}\).
• The product of the roots \(x_1 \cdot x_2 = \frac{c}{a}\).

In the case of the equation \(-2x^2 + 4x - 3 = 0\), the calculated roots are verified using these formulas:

• Sum of roots: \(1 - \frac{i\sqrt{2}}{2} + 1 + \frac{i\sqrt{2}}{2} = 2\), aligning with \(\frac{-4}{-2} = 2\).
• Product of roots: \((1 - \frac{i\sqrt{2}}{2})(1 + \frac{i\sqrt{2}}{2}) = 1 + \frac{2}{4} = 1.5\), matching \(\frac{-3}{-2} = 1.5\).

This verification not only confirms the correctness of the roots but also emphasizes the mathematical consistency in quadratic equations.