Question

Use the Quadratic Formula to solve the quadratic equation. $$ 36 x^{2}+24 x=7 $$

Short answer

The solutions to the equation are \(x = 0.1\) and \(x = -1.94\)

Step-by-step solution

Rewrite in standard form

Rewrite the equation in the standard form of \(ax^{2} + bx + c = 0\). That gives \(36x^{2} + 24x - 7 = 0\).

Identify coefficients

From the standard form, it can be seen that \(a = 36\), \(b = 24\), and \(c = -7\).

Apply the Quadratic Formula

Applying the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) with the coefficients gives two possible solutions for x: \(x = \frac{-24 \pm \sqrt{24^{2} - 4 * 36 * -7}}{2 * 36}\) .

Simplify the solution

Solving the equation inside the square root, and simplifying the equation gives two possible solutions: \(x = \frac{-24 + \sqrt{576 + 1008}}{72}\) and \(x = \frac{-24 - \sqrt{576 + 1008}}{72}\). After further simplification, we get \(x = 0.1\) and \(x = - 1.94\)

Key concepts

Quadratic Formula

The Quadratic Formula is a powerful tool used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula applies to quadratics, equations that form a parabolic curve when graphed. To solve these types of equations, the formula given is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). To use this formula, you need to identify the coefficients \(a\), \(b\), and \(c\) from your equation. These coefficients are vital as they fit into specific parts of the quadratic formula, allowing you to solve for \(x\). Once you substitute the values, you perform arithmetic inside the square root and then simplify your answer. This can yield one or two possible solutions, because of the \(\pm\) which gives both a positive and negative outcome. Remember that handling the square root properly and ensuring all steps of simplification are completed leads to finding the correct solutions.

Standard Form

Standard Form is the way we arrange quadratic equations to make them easier to solve. The format is \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) represent constants. This arrangement helps easily identify the coefficients necessary for the Quadratic Formula. The main goal is to move all terms involving \(x\) and the constant to one side of the equation, ensuring the other side is zero. For example, if you start with \(36x^2 + 24x = 7\), rearranging it into \(36x^2 + 24x - 7 = 0\) puts it in standard form. This allows for a direct application into formulas or methods, keeping all components aligned clearly and facilitating simple identification of all necessary parts for calculation.

Coefficients

In quadratic equations, coefficients are the numbers in front of the variables that affect the parabola's shape and position. Every quadratic equation has three coefficients: \(a\), \(b\), and \(c\).- \(a\) is the coefficient of \(x^2\), influencing the opening direction and width of the parabola. - \(b\) is the coefficient of \(x\), affecting the left-right positioning of the parabola. - \(c\) is the constant term, which determines the point where the parabola crosses the y-axis.For the equation \(36x^2 + 24x - 7 = 0\), the coefficients are \(a = 36\), \(b = 24\), and \(c = -7\). Correctly identifying these values is crucial when using the Quadratic Formula, as even a small mistake in this can lead to incorrect solutions. Understanding these components lets you know exactly how the graph will look and react to changes in the variables.

Arithmetic Simplification

Arithmetic simplification involves performing mathematical operations to reduce and simplify your equation or solution. This process takes initial, often complex, results and condenses them into a simpler, more digestible answer.When using the Quadratic Formula, simplification involves:

• Calculating values inside the square root (discriminant), such as \(b^2 - 4ac\).
• Simplifying the results from arithmetic operations, such as addition, subtraction, multiplication, or division.
• Arriving at neat solutions that can be fractions, integers, or even square roots.

In our example, the expression \(\sqrt{576 + 1008}\) simplifies further to find two roots: \(x = 0.1\) and \(x = -1.94\). Each step is imperative to hone in on accuracy, breaking down complex steps to ensure the integrity of your final solutions. Thus, mastering arithmetic simplification enables precise and efficient solving of quadratic equations.