Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 4 x^{2}+4 x=7 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \( 4x^{2} + 4x - 7 = 0 \) are \( x = 0.9142 \) and \( x = -1.9142 \) respectively.

Step 1: Identifying the coefficients

This is a quadratic equation, and we can identify the coefficients. Coefficient \( a = 4 \), \( b = 4 \), and \( c = -7 \).

Step 2: Applying the Quadratic Formula

Use the quadratic formula which is \( x = (-b ± \sqrt{b^2 - 4ac}) / 2a \). Plugging the values of a, b and c into the formula, we get \( x = (-4 ± \sqrt{4^2 - 4*4*-7}) / 2*4 \)

Step 3: Simplifying the equation

Simplify the equation to get the solutions. We get \( x = (-4 ± \sqrt{16 + 112}) / 8 = (-4 ± \sqrt{128}) / 8 \). This simplifies further to \( x = -0.5 ± 1.4142 \).

Step 4: Final Solution

So the solutions are \( x = -0.5 + 1.4142 =0.9142 \) and \( x = -0.5 - 1.4142 = -1.9142 \).

Key Concepts

Quadratic Equation

A quadratic equation is an algebraic expression set in the form \( ax^2 + bx + c = 0 \).
This equation represents a curve on a graph known as a parabola.
Quadratic equations are fundamental in algebra because they model various real-world situations, like projectile motion or area calculations.
The degree of the equation, indicated by the highest power of the variable \( x \) (which is 2 in this case), categorizes it as a quadratic.
Understanding the quadratic equation is crucial for solving many mathematical problems.
Whether through factorization or formulas, different methods are applied based on the specifics of the equation at hand.

Coefficients Identification

In solving any quadratic equation using the quadratic formula, the first step is to identify the coefficients: \( a \), \( b \), and \( c \).
These coefficients are vital because they appear in almost every step of the solving process.
The standard form of the quadratic equation is \( ax^2 + bx + c = 0 \).

• For the equation given in the exercise \( 4x^2 + 4x = 7 \), we first rearrange it to get \( 4x^2 + 4x - 7 = 0 \).
• Here, \( a = 4 \), \( b = 4 \), and \( c = -7 \).

Recognizing these coefficients accurately is crucial for applying the quadratic formula successfully.

Solution Simplification

After finding values through the quadratic formula, simplifying the obtained expression leads to the final solution.
Following our exercise, after applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), the solved parameters give us \( x = \frac{-4 \pm \sqrt{128}}{8} \).
Simplifying \( \sqrt{128} \) can further improve the understanding and clarity of the solution:

• \( \sqrt{128} = 8\sqrt{2} \), which simplifies calculations further.
• Thus, the solutions become more precise numerically.

This simplification yields \( x = -0.5 \pm 1.4142 \), ensuring our answers are easier to interpret.

Algebra

Algebra provides the foundational operations and symbols essential to tackle quadratic equations.
It involves the systematic manipulation of symbols and numbers to solve for unknowns.

• In our context, the quadratic formula itself is an algebraic tool derived from rearranging the quadratic equation.
• Through substitution and elimination, we reach the roots or solutions which satisfy the equation.
• This process emphasizes algebra's versatility in solving various types of equations, beyond just quadratics.

Mastering these basic algebraic principles makes math more manageable and paves the way for tackling even more complex problems in the future.