Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ 4 x-7=x^{2} $$

Step-by-Step Solution

Answer

The imaginary roots are \( x = 2 \pm i \sqrt{3} \).

Step 1: Arrange into Standard Form

Start by rearranging the given equation into the standard quadratic form \( ax^2 + bx + c = 0 \). The given equation is \( 4x - 7 = x^2 \). Rearrange it to \( x^2 - 4x + 7 = 0 \).

Step 2: Identify Coefficients

Identify the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -4 \), and \( c = 7 \).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for the roots. Substitute \( a = 1 \), \( b = -4 \), and \( c = 7 \) into the formula.

Step 4: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \) where \( b = -4 \), \( a = 1 \), and \( c = 7 \). The discriminant is \( (-4)^2 - 4 \cdot 1 \cdot 7 = 16 - 28 = -12 \).

Step 5: Find the Roots

Since the discriminant is negative, the roots are imaginary. Substitute back into the quadratic formula: \( x = \frac{-(-4) \pm \sqrt{-12}}{2 \cdot 1} \). Simplify to \( x = \frac{4 \pm \sqrt{-12}}{2} = \frac{4 \pm i \sqrt{12}}{2} = \frac{4 \pm 2i\sqrt{3}}{2} \), which simplifies further to \( x = 2 \pm i\sqrt{3} \).

Key Concepts

Imaginary Roots

When solving quadratic equations, sometimes you end up with a negative number under the square root in the quadratic formula. This happens when the discriminant, which is the part of the formula under the square root (i.e., \(b^2 - 4ac\)), is negative.
Negative discriminants indicate that the equation has no real roots, only imaginary roots.
Imaginary numbers are expressed using the imaginary unit \(i\), where \(i = \sqrt{-1}\). This allows us to handle the square root of a negative number.

• If the discriminant is negative, the roots will be complex numbers, expressed as \(a \, \pm \, bi\).
• In our example, we calculated the roots as \(2 \, \pm \, i\sqrt{3}\), implying that 2 is the real part, and \(i\sqrt{3}\) is the imaginary part.

Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is found using the formula \(b^2 - 4ac\).
The discriminant helps determine the nature of the roots of the quadratic equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the quadratic equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is less than zero, as in our example, the equation has two complex conjugate imaginary roots.

In the provided problem, the discriminant was \(-12\) meaning the roots of the equation are imaginary.

Standard Form

The standard form of a quadratic equation is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
This form is crucial because it allows us to identify the coefficients that we need to use in the quadratic formula. The equation must be set to equal zero to be in standard form.

• In our example, the original equation was \(4x - 7 = x^2\).
• To convert it to standard form, we rearranged it to \(x^2 - 4x + 7 = 0\).
• This made identifying \(a = 1\), \(b = -4\), and \(c = 7\) straightforward, facilitating further calculations.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \(x\), of the form \(ax^2 + bx + c = 0\), where "a" is not equal to zero.
Quadratic equations are called so because "quad" means square, indicating that the highest power of \(x\) is 2.

• Every quadratic equation can have zero, one, or two real solutions depending on the discriminant.
• They can also have two complex solutions, which happen when the discriminant is negative, as we have seen.
• The solutions are found using the quadratic formula, factoring, or completing the square, depending on the problem at hand.

By using these methods, we can effectively solve for the values of \(x\) that satisfy the equation.