Problem 18 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. Then use the quadratic formula to solve the equation. $$4 x^{2}+4 x=-1$$

Step-by-Step Solution

Answer

The solution to the equation \(4x^{2} + 4x + 1 = 0\) is \(x = - \frac{1}{2} \)

Step 1: Convert the equation to standard form

Rearrange the given equation \(4x^{2} + 4x + 1 = 0\) to make it in standard form.

Step 2: Identify a, b, c from the equation

In this equation, a is the coefficient of \(x^{2}\), b is the coefficient of x, and c is the constant. So, here a = 4, b = 4, c = 1.

Step 3: Calculate the Discriminant

Discriminant, D is given by \(D = b^{2} - 4ac\). Substituting the values of a, b and c, we get \(D = (4)^{2} - 4*4*1 = 16 - 16 = 0\).

Step 4: Use the quadratic formula

The roots of the equation can be found using the quadratic formula, which is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substituting the values of a, b and D, we get \(x = \frac{-4 \pm \sqrt{0}}{2*4} = - \frac{1}{2}\).

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation, which is expressed as
\( ax^2 + bx + c = 0 \), is key to solving quadratics. Here, \( a \), \( b \), and \( c \) represent known numbers where \( a \) is not zero, and \( x \) is the variable. The equation must equal zero to be in standard form.

In the exercise \( 4x^2 + 4x = -1 \), we begin by moving all terms to one side to achieve standard form, resulting in \( 4x^2 + 4x + 1 = 0 \). The coefficients are then easily identifiable: \( a=4 \), \( b=4 \), and \( c=1 \). Recognizing the standard form allows us to tackle the quadratic using various methods, such as factoring, completing the square, or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations and is applicable when other methods are complex or unfeasible. It is given as
\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). To use this formula, we substitute the coefficients \( a \), \( b \), and \( c \) from the standard form of the equation.

Applied to the exercise problem, we plug in \( a=4 \), \( b=4 \), and \( c=1 \). The quadratic formula ensures we find both possible solutions for \( x \), which represent the points where the parabola crosses the x-axis. When the discriminant, the value under the square root, is zero, it indicates a unique solution, as seen in the exercise where the result simplifies to \( x = -1/2 \).

Discriminant

The discriminant is the part of the quadratic formula under the square root, denoted as \( D \). It is calculated with the expression \( D = b^2 - 4ac \).

The discriminant tells us about the nature of the roots:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root (also known as a repeated or double root).
• If \( D < 0 \), there are no real roots, but two complex roots.

For the given exercise, after identifying \( a=4 \), \( b=4 \), and \( c=1 \), we calculate the discriminant as \( D = (4)^2 - 4 \cdot 4 \cdot 1 = 16 - 16 = 0 \), indicating one real solution. It reinforces the result from the quadratic formula and completes our understanding of the equation's roots.