Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$2 x^{2}=4 x+30$$

Short answer

The solutions to the equation are \(x = 5\) or \(x = -3\).

Step-by-step solution

Rewrite the equation in standard form

First, the equation needs to be rearranged such that it is equal to zero. Subtracts \(4x\) and 30 from both sides to get \(2x^2 - 4x - 30 = 0\).

Apply the Quadratic Formula

In the formula \(ax^2 + bx + c = 0\), there is \(a = 2\), \(b = -4\), and \(c = -30\). Substitute these values into the quadratic formula to calculate \(x\): \(x = \frac{-(-4) ± \sqrt{(-4)^2 - 4*2*(-30)}}{2 * 2}\). Simplifying this gives \(x = \frac{4 ± \sqrt{16 + 240}}{4}\) which reduces to \(x = \frac{4 ± \sqrt{256}}{4}\).

Solve for x

Simplify the equation further to find the two possible values of \(x\). \(x = \frac{4 ± 16}{4}\) gives \(x = 5\) or \(x = -3\).

Key concepts

Standard Form of Quadratic Equation

Quadratic equations are second-degree polynomial equations that describe a parabola when graphed. The standard form of a quadratic equation is represented as \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the variable. The coefficient \(a\) must be nonzero; if it were zero, the equation would not be quadratic but a linear one. In our given exercise, the original equation \((2x^2 = 4x + 30)\) is not in standard form. To convert it to standard form, one must rearrange the terms to get \(2x^2 - 4x - 30 = 0\), which follows the \(ax^2 + bx + c = 0\) format.

It is crucial in algebra to be able to identify and rearrange equations into this standard form because it is the starting point for many methods of solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The process of converting to standard form requires operations like addition or subtraction to both sides of the equation to consolidate all terms on one side and set the equation to equal zero.

Solving Quadratic Equations

Once in standard form, there are several ways to solve a quadratic equation, and one of the most universally applicable methods is the quadratic formula. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides the solutions by finding the roots of the quadratic equation directly from the coefficients \(a\), \(b\), and \(c\).

To use this formula, simply identify and substitute the values of the coefficients from the standard form of the equation into the formula and perform the arithmetic operations. In our example, we substitute \(a=2\), \(b=-4\), and \(c=-30\) into the quadratic formula, then calculate the discriminant \(b^2 - 4ac\), and finally, divide by \(2a\) to determine the potential values for \(x\). The quadratic formula will always find the roots, whether they are real or complex, making it a foolproof method as long as the equation is in the correct form and the arithmetic is carried out correctly.

Quadratic Equation Roots

The roots of a quadratic equation, also known as the solutions or zeros, represent the values of \(x\) where the quadratic graph crosses the x-axis. These roots can be real or complex numbers and there can be one (in the case of a perfect square) or two different roots. When using the quadratic formula, the term under the square root, \(b^2 - 4ac\), is known as the discriminant. The discriminant determines the nature of the roots:

• If \(b^2 - 4ac > 0\), two distinct real roots exist.
• If \(b^2 - 4ac = 0\), one real root exists, which is a repeated root.
• If \(b^2 - 4ac < 0\), two complex roots exist.

The discriminant provides a quick way to predict the outcome without fully solving the equation. For our exercise, the discriminant is \(16 + 240\), which is a positive number indicating two real roots. Simplifying the formula gives us two solutions for \(x\): \(5\) and \( -3\). These two numbers are the points where the graph of our quadratic equation would intersect with the x-axis.