Problem 13 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$2 x^{2}-4 x+7=0$$

Step-by-Step Solution

Answer

The equation has no real solutions, but two complex solutions \( x = 1 \pm \frac{\sqrt{10}}{2}i \).

Step 1: Identify the Type of Equation

The given equation is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -4 \), and \( c = 7 \).

Step 2: Calculate the Discriminant

The discriminant \( \, \Delta \, \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). Calculate it as follows: \( (-4)^2 - 4 \cdot 2 \cdot 7 = 16 - 56 = -40 \).

Step 3: Analyze the Discriminant

Since the discriminant \( \Delta = -40 \) is less than zero, the quadratic equation has no real solutions but two complex solutions.

Step 4: Use the Quadratic Formula to Find Solutions

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substituting in the values, we have: \( x = \frac{-(-4) \pm \sqrt{-40}}{4} \).

Step 5: Simplify the Complex Solutions

First, simplify the expression under the square root (which is the imaginary part): \( \sqrt{-40} = \sqrt{40}i = 2\sqrt{10}i \). Thus, the solutions are: \( x = \frac{4 \pm 2\sqrt{10}i}{4} \).

Step 6: Finalize the Solution

Simplify the solutions to get: \( x = 1 \pm \frac{\sqrt{10}}{2}i \). The solutions to the equation \( 2x^2 - 4x + 7 = 0 \) are \( x = 1 + \frac{\sqrt{10}}{2}i \) and \( x = 1 - \frac{\sqrt{10}}{2}i \).

Key Concepts

Discriminant

The discriminant is a key part of solving quadratic equations. It's a formula that tells us a lot about the nature of the solutions for these equations. The discriminant comes from the general quadratic equation \( ax^2 + bx + c = 0 \). This simple expression is calculated as \( \Delta = b^2 - 4ac \). How does it help?

• If the discriminant is positive, there are two different real solutions.
• If it is zero, there's exactly one real solution.
• If it is negative, like in our exercise where \( \Delta = -40 \), we have no real solutions, but instead two complex ones.

This simple calculation saves a lot of time by allowing us to know right away what type of solutions we can expect when solving the quadratic equation.

Complex Solutions

When the discriminant is negative, this indicates that the solutions to the quadratic equation are complex. But what does it mean for a solution to be complex?Complex numbers include a real part and an imaginary part. The imaginary unit, denoted as \( i \), has the property that \( i^2 = -1 \). Thus when we encounter solutions involving the square root of a negative number, we express them in terms of \( i \).In our original exercise, the solution involves squaring \(-40\), which becomes \( \sqrt{-40} = 2\sqrt{10}i \). The two solutions will be \( 1 + \frac{\sqrt{10}}{2}i \) and \( 1 - \frac{\sqrt{10}}{2}i \). Both of these solutions are in the form \( a + bi \), representing complex numbers, where \( a \) is the real part and \( bi \) is the imaginary part.

Quadratic Formula

The Quadratic Formula is a powerful tool for solving any quadratic equation, even those with complex solutions. The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides a straightforward way to find solutions by simply substituting the values of \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \).

• The term \( -b \) gives the opposite sign of \( b \) and acts as a starting point for both solutions.
• The expression under the square root, \( b^2 - 4ac \), is the discriminant which we've covered earlier.
• The "\( \pm \)" symbols indicate two potential solutions: one for addition and one for subtraction.
• Finally, division by \( 2a \) helps in scaling the result back to the level needed for solving the equation.

Using this formula, our exercise transformed directly into the solutions \( x = 1 + \frac{\sqrt{10}}{2}i \) and \( x = 1 - \frac{\sqrt{10}}{2}i \), incorporating both real and imaginary components.