Problem 81 - Elementary Algebra

Question

Solve using the quadratic formula. $$ (2 x+3) 2=16 x+4 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2} + i\) and \(x = \frac{1}{2} - i\).

Step 1: Expand the Equation

Start by expanding the left side of the equation. The equation given is \((2x + 3)^2 = 16x + 4\). To expand \((2x + 3)^2\), use the formula \((a+b)^2 = a^2 + 2ab + b^2\). Thus:\[(2x+3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9\]Now the equation becomes \(4x^2 + 12x + 9 = 16x + 4\).

Step 2: Rearrange the Equation

Move all terms to one side to form a standard quadratic equation \(ax^2 + bx + c = 0\). Start by subtracting \(16x\) and \(4\) from both sides:\[4x^2 + 12x + 9 - 16x - 4 = 0\]Simplify the terms to get:\[4x^2 - 4x + 5 = 0\]

Step 3: Identify Coefficients

Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation \(4x^2 - 4x + 5 = 0\). Here, \(a = 4\), \(b = -4\), and \(c = 5\).

Step 4: Apply the Quadratic Formula

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the values of \(a\), \(b\), and \(c\) into the formula:\[x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 4 \times 5}}{2 \times 4}\]

Step 5: Simplify the Expression

Calculate the discriminant and simplify:\[x = \frac{4 \pm \sqrt{16 - 80}}{8}\]The discriminant \(16 - 80 = -64\), thus:\[x = \frac{4 \pm \sqrt{-64}}{8}\]

Step 6: Solve for Complex Roots

Since the discriminant \(-64\) is negative, the roots are complex. Recall that \(\sqrt{-1} = i\), so \(\sqrt{-64} = 8i\). Therefore:\[x = \frac{4 \pm 8i}{8}\]Separate the terms:\[x = \frac{4}{8} \pm \frac{8i}{8}\]This simplifies to:\[x = \frac{1}{2} \pm i\]

Step 7: Conclusion

The solutions to the quadratic equation \(4x^2 - 4x + 5 = 0\) using the quadratic formula are \(x = \frac{1}{2} + i\) and \(x = \frac{1}{2} - i\).

Key Concepts

Complex Roots

Sometimes, when we solve a quadratic equation, the roots aren’t real numbers but instead complex numbers. This happens when the equation has a negative discriminant, which we will discuss more in the next section. When this occurs, the roots involve the imaginary unit, denoted as \( i \), where \( i = \sqrt{-1} \).
In the example quadratic equation \(4x^2 - 4x + 5 = 0\), we ended up with complex roots. Because the discriminant was negative, we found ourselves with \(\sqrt{-64}\), which equals \(8i\). Thus, the solutions were \(x = \frac{1}{2} + i\) and \(x = \frac{1}{2} - i\).

• The term \(\pm 8i\) indicates that these complex roots are conjugates, a common pattern in quadratic equations with complex solutions.
• Complex numbers combine a real part and an imaginary part, like \(x = \frac{1}{2} \pm i\). This helps in maintaining the symmetry often found in polynomial roots, even when they are not real.

Discriminant

The discriminant is a key part of the quadratic formula and helps us determine the nature of the roots of a quadratic equation. The discriminant is part of the expression under the square root in the quadratic formula, \( b^2 - 4ac \).

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, known also as a repeated root.
• When the discriminant is negative, as in our example, it indicates that the quadratic equation has complex roots.

By calculating the discriminant of our quadratic equation \(4x^2 - 4x + 5 = 0\), we found it to be \(-64\), which confirmed the presence of complex roots. This is why we ended up with \(x = \frac{1}{2} \pm i\) in the solution.
Understanding the discriminant makes it easier to predict the type of solutions without solving the entire equation.

Quadratic Equation

A quadratic equation is any equation that can be rearranged into the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents an unknown. The highest exponent here is \( x^2 \), which makes it 'quadratic.'
In our problem, we started with \((2x+3)^2 = 16x + 4\). To bring this into standard form, we expanded and rearranged the terms to achieve \(4x^2 - 4x + 5 = 0\).

• The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is an efficient tool to solve any quadratic equation. It directly provides the solutions, either real or complex.
• By identifying coefficients \( a = 4 \), \( b = -4 \), and \( c = 5 \), and using these in the quadratic formula, we solve for \(x\) quickly.
• Quadratic equations appear frequently in various fields such as physics, engineering, and economics, showing their practical importance beyond mathematics.