Problem 52 - Algebra 2

Question

Solve each equation by factoring, by taking square roots, or by graphing. If necessary, round your answer to the nearest hundredth. $$ 2 x^{2}-6 x=8 $$

Step-by-Step Solution

Answer

The solutions to the equation \(2x^2 - 6x - 8 = 0\) are \(x = 4\) and \(x = -1\).

Step 1: Subtract 8 from both sides

To set the equation to zero, which is necessary for solving by factoring or taking square roots, subtract 8 from both sides of the equation. This will give you the quadratic equation in standard form.

Step 2: Rewrite the equation

After subtracting 8 from both sides, the equation becomes: \(2x^2 - 6x - 8 = 0\). Now, it's ready to be factored or solved using square roots.

Step 3: Factor the quadratic

Factor the quadratic equation \(2x^2 - 6x - 8\) to find the values of \(x\) that make the equation true. If the quadratic does not factor easily, you can use the quadratic formula to find the roots. For factoring, look for two numbers that multiply to give \(-16\) (the product of the coefficient of \(x^2\) and the constant term) and add to give \(-6\), the coefficient of \(x\).

Step 4: Solve the factored equation

Set each of the factored expressions equal to zero and solve for \(x\). If you're unable to factor the equation, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -6\), and \(c = -8\).

Step 5: Simplify the solutions

After solving the factored expressions, simplify the solutions to the equation. If the problem asks for solutions to the nearest hundredth, round your answers as necessary.

Key Concepts

Factoring Quadratics

When faced with a quadratic equation like \(2x^2 - 6x = 8\), one of the most straightforward methods to solve it is factoring. Factoring transforms the quadratic equation into a product of binomials or trinomials. Simplifying a quadratic into factors can make finding the roots much easier as you will be setting each factor equal to zero to solve for the variable \(x\).

For the given equation, we first rewrite it in standard form as \(2x^2 - 6x - 8 = 0\). Then, we look for two numbers that multiply to the product of the coefficient of \(x^2\) and the constant term (in this case, -16), and add up to the coefficient of \(x\) (which is -6). Once the factors are found, they are set in a form such as \((2x + m)(x + n) = 0\), where \(m\) and \(n\) are the found numbers. We can then solve for \(x\) by setting each factor to zero. This method is often taught in algebra classes as it lays a strong foundation for understanding the behavior of quadratic functions.

Quadratic Formula

In cases where factoring is too complex or not possible, the quadratic formula is a reliable tool. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). It is derived from the process of completing the square and provides the roots directly.

For the equation \(2x^2 -6x - 8 = 0\), we identify \(a = 2\), \(b = -6\), and \(c = -8\). Plugging these values into the quadratic formula allows for the calculation of the roots. This process includes finding the discriminant (\(b^2 - 4ac\)), taking its square root, and applying the formula to find the two potential values of \(x\). The quadratic formula is a powerful technique that guarantees a solution for any quadratic equation, provided the discriminant is not negative (which would lead to complex solutions).

Taking Square Roots

For certain quadratics, especially those in the form \(x^2 = k\), taking square roots is a simple and efficient method to find the roots. When an equation can be rearranged in such a way that \(x^2\) is isolated on one side, taking the square root of both sides of the equation will give us \(x = \pm\sqrt{k}\).

Applying the Method

However, our example \(2x^2 - 6x = 8\) does not fit this form perfectly as is. First, we would need to complete the step of setting the equation to zero and attempt to simplify it so that the \(x^2\) term is isolated. If we could do so, we would then apply the square root to both sides, remember to consider both the positive and negative square roots, and solve for \(x\). This method is limited to certain types of quadratic equations and is not as universally applicable as factoring or the quadratic formula.

Graphing Quadratics

Graphing is a visual method of solving quadratic equations and understanding the function's properties. By graphing the quadratic function, we can easily identify the roots (also known as zeros or x-intercepts), the vertex, and the axis of symmetry.

For graphing the quadratic equation \(2x^2 - 6x - 8 = 0\), we use its standard form to find key points and plot them on a coordinate plane. The solutions to the equation correspond to the points where the graph crosses the x-axis. It is possible to estimate these roots by observing the graph or using graphing technology for a more precise calculation.

Advantages of Graphing

Graphing not only provides the solutions but also a comprehensive look at the behavior of the quadratic function over different intervals. It is especially useful when dealing with quadratic equations that do not produce nice, whole number solutions, where estimating to the nearest hundredth might be necessary, as indicated in the given problem's instructions.