Problem 57 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate. $$ x^{2}+12 x=4 $$

Step-by-Step Solution

Answer

The solutions are \( x = -6 + 2\sqrt{10} \) and \( x = -6 - 2\sqrt{10} \).

Step 1: Rearrange the Equation

Start by moving all terms to one side of the equation. Subtract 4 from both sides to set the equation equal to zero. This gives us:\[ x^2 + 12x - 4 = 0 \]

Step 2: Identify Coefficients for the Quadratic Formula

Given the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients:- \( a = 1 \)- \( b = 12 \)- \( c = -4 \)

Step 3: Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \) to find the roots of the equation. Substitute the identified values:\[ x = \frac{{-12 \pm \sqrt{{12^2 - 4(1)(-4)}}}}{2(1)} \]

Step 4: Compute the Discriminant

Calculate the discriminant \( b^2 - 4ac \):\[ 12^2 - 4(1)(-4) = 144 + 16 = 160 \]The discriminant is 160.

Step 5: Calculate the Roots

Substitute the discriminant back into the quadratic formula:\[ x = \frac{{-12 \pm \sqrt{160}}}{2} \]Simplify \( \sqrt{160} \) to \( 4\sqrt{10} \). Thus:\[ x = \frac{{-12 \pm 4\sqrt{10}}}{2} \]Further simplify:\[ x = -6 \pm 2\sqrt{10} \]

Step 6: Present the Solution

The solutions to the quadratic equation \( x^2 + 12x - 4 = 0 \) are:\[ x = -6 + 2\sqrt{10} \]\[ x = -6 - 2\sqrt{10} \]

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to calculate the roots, even when the equation does not factor easily.
The formula is:

• \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)

The symbols "\( \pm \)" indicate that there are generally two possible solutions for \( x \). Applying this formula requires determining the coefficients \( a \), \( b \), and \( c \) from the equation, and plugging them into the formula to find the possible values of \( x \). It's especially handy for solving equations like \( x^2 + 12x - 4 = 0 \), where simple factoring is not obvious.
The beauty of the quadratic formula lies in its ability to handle any kind of quadratic equation and directly involve the discriminant to help determine the types of roots.

Discriminant

The discriminant in a quadratic equation is an essential component. It is part of the quadratic formula and is given by the expression \( b^2 - 4ac \).
This value serves a crucial role by revealing the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, which means the roots are repeated.
• If the discriminant is negative, there are no real roots, but two complex roots.

The discriminant is like a detective in the quadratic equation world, allowing us to understand more about the solutions before we even calculate them. In the equation \( x^2 + 12x - 4 = 0 \), the discriminant is calculated as \( b^2 - 4ac = 160 \), indicating two distinct real roots. Understanding this part of the formula can help in predicting the outcome of your solutions before performing more calculations.

Roots of a Polynomial

The roots of a polynomial, particularly a quadratic one, are the solutions where the polynomial equals zero. In simpler terms, they are the values of \( x \) when the equation \( ax^2 + bx + c = 0 \) is satisfied.
These roots can be found using different methods, such as:

• Factoring (when possible)
• Completing the square
• Using the Quadratic Formula

In our example, \( x^2 + 12x - 4 = 0 \), we used the quadratic formula to find that the roots are \( x = -6 + 2\sqrt{10} \) and \( x = -6 - 2\sqrt{10} \). These roots can also be thought of as the x-intercepts of the polynomial graph, where the curve crosses the x-axis. Knowing and calculating the roots helps us understand the behavior of the polynomial function.

Algebraic Solutions

Finding algebraic solutions means solving the equation using algebraic methods to express the solutions in a precise form.
The quadratic equation \( ax^2 + bx + c = 0 \), as seen with \( x^2 + 12x - 4 = 0 \), can be tricky because it might not easily factorize into a product of binomials. This is where algebraic solutions shine, providing a path to exact answers without numerical approximation.The quadratic formula plays a key role here, offering an exact algebraic solution even if the equation can't be simply factored. The solutions we obtained, \( x = -6 + 2\sqrt{10} \) and \( x = -6 - 2\sqrt{10} \), are complete algebraic expressions, presenting the answer as a radical solution if it does not simplify to an integer or simple fraction. Algebraic solutions are preferred when precision is required, especially in academic and professional contexts.