Problem 50 - Algebra 2

Question

Solve each equation by factoring, by taking square roots, or by graphing. If necessary, round your answer to the nearest hundredth. $$ 4 x^{2}+4 x=3 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{3}{2}\).

Step 1: Rewrite the equation in standard form

To solve the quadratic equation by factoring or taking square roots, you first need to rewrite it in the standard form, which is \(ax^2 + bx + c = 0\). Subtract 3 from both sides to get the equation in standard form: \(4x^2 + 4x - 3 = 0\).

Step 2: Factor the quadratic equation

In this case, the quadratic equation \(4x^2 + 4x - 3\) does not factor easily. Therefore, we will use the quadratic formula to find the solutions.

Step 3: Apply the quadratic formula

The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). For the equation \(4x^2 + 4x - 3 = 0\), \(a = 4\), \(b = 4\), and \(c = -3\). Plug in these values and solve for \(x\).

Step 4: Solve for x

Plug in the values into the quadratic formula:\[ x = \frac{{-4 \pm \sqrt{{(4)^2 - 4 \cdot 4 \cdot (-3)}}}}{{2 \cdot 4}} = \frac{{-4 \pm \sqrt{{16 + 48}}}}{{8}} = \frac{{-4 \pm \sqrt{{64}}}}{8} \]Since \(\sqrt{64} = 8\), the solutions are:\[ x = \frac{{-4 + 8}}{8} = \frac{1}{2} \] and \[ x = \frac{{-4 - 8}}{8} = -\frac{3}{2} \].

Step 5: Round the answers (if necessary)

The solutions calculated are \(\frac{1}{2}\) and \(-\frac{3}{2}\), and they are already in the simplest form. Rounding is not necessary since these are exact values.

Key Concepts

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation as a product of two or more simpler expressions. It's a fundamental technique used to solve quadratics, and it is often the first method taught because of its straightforward approach. A quadratic equation is in the form of \(ax^{2} + bx + c = 0\), and the goal is to find two binomials that when multiplied together, give the original quadratic.

For example, if we have \(x^{2} + 5x + 6 = 0\), we can factor it into \((x + 2)(x + 3) = 0\). By setting each factor equal to zero, we can solve for the roots of the equation, which in this case are \(x = -2\) and \(x = -3\).

However, sometimes quadratics cannot be easily factored, as seen in the original exercise. In such cases, other methods such as the quadratic formula or graphing need to be employed. Moreover, factoring requires practice and a keen eye for identifying common factors and patterns.

Quadratic Formula

When factoring can be complex or impossible, the quadratic formula is a reliable tool. It is expressed as \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) and can solve any quadratic equation in the form of \(ax^{2} + bx + c = 0\). The formula offers a systematic way to find the roots of the equation by replacing the coefficients \(a\), \(b\), and \(c\) from the equation.

In the exercise provided, the quadratic formula was used due to the difficulty in factoring the equation \(4x^2 + 4x - 3 = 0\). The solutions are found by calculating the discriminant, which is the part under the square root, \(\sqrt{{b^2 - 4ac}}\). The discriminant provides valuable information; a positive discriminant indicates two real solutions, zero means one real solution, and a negative discriminant would have no real solutions, but rather complex ones. The quadratic formula is a robust method, offering solutions regardless of the nature of the discriminant.

Taking Square Roots

Taking square roots is another approach to solve certain types of quadratic equations, specifically those that can be simplified into the form \(x^{2} = d\), where \(d\) is some constant. When an equation is rewritten this way, you can easily solve for \(x\) by taking the square root of both sides.

For instance, if you have an equation like \(x^{2} = 16\), you would take the square root of 16 to find that \(x\) could be \(+4\) or \(-4\) because both numbers squared will return 16. Remember that whenever you take a square root in an equation, you must consider both the positive and negative solutions since they both are valid for the square of a number. However, this method is limited to equations that can be manipulated into this simplified form and won't work for all quadratic equations.

Graphing Quadratic Equations

Graphing quadratic equations involves plotting the curve of a quadratic function on a coordinate axis. Each function produces a parabola, a symmetrical curve with a highest or lowest point known as the vertex. To graph a function, we use the standard quadratic form \(y = ax^{2} + bx + c\). The values of \(a\), \(b\), and \(c\) determine the shape and position of the parabola.

By graphing, you can visually find the x-intercepts, which correspond to the solutions of the quadratic equation. These are the points where the parabola crosses the x-axis. These solutions are also known as the roots or zeroes of the function. For example, if a graph shows the parabola crossing the x-axis at points \(1\) and \(-3\), those are the solutions to the quadratic equation. Graphing can be useful not only for finding the roots but also for understanding the behavior of the equation, including its direction (upward or downward) and the vertex, which reflects the equation's maximum or minimum value.