Problem 68 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. (Lesson 9.6) $$2 x^{2}+4 x=7$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{-4 + \sqrt{72}}{4}\) and \(x = \frac{-4 - \sqrt{72}}{4}\)

Step 1: Rewrite the equation in standard form

To set this equation to standard form, subtract 7 from both sides to get:\[2x^{2} + 4x - 7 = 0\]

Step 2: Apply the Quadratic Formula

The Quadratic Formula is \(x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). In our equation, a is 2, b is 4, and c is -7. Substitute these values into the Quadratic Formula to get:\[x=\frac{-4 \pm \sqrt{(4)^{2}-4 \cdot 2 \cdot (-7)}}{2 \cdot 2}\]

Step 3: Simplify the expression

Doing the math within the square root first, we find that \(b^{2}-4ac = 16 - (-56) = 72\). Thus, we get \[x=\frac{-4 \pm \sqrt{72}}{4}\]

Step 4: Get the final solution

Finally, simplifying the above gives us the roots of the equation:\[x = \frac{-4 + \sqrt{72}}{4} \quad or \quad x = \frac{-4 - \sqrt{72}}{4}\]

Key Concepts

Quadratic Formula

The quadratic formula is a must-know tool for solving any quadratic equation. It is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a\), \(b\), and \(c\) represent the coefficients from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).
Using this formula allows us to directly find the roots of the quadratic equation without the need for factoring or graphing.

• The term \(b^2 - 4ac\) is known as the "discriminant". It helps determine the number and type of roots.
• If the discriminant is positive, the equation has two distinct real roots.
• If it's zero, there is one real root (a repeated root).
• If the discriminant is negative, the quadratic equation has two complex roots.

Understanding the discriminant gives insight into what type of solutions you can expect even before solving the equation. So, next time you encounter a quadratic equation, remember the quadratic formula and the discriminant!

Standard Form

In order to effectively use the quadratic formula, your quadratic equation must be in "standard form". This standard form is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.
When given a quadratic equation, your first task is to rearrange it into this format. For example, in the equation \(2x^2 + 4x = 7\), we needed to move all terms to one side of the equation to get \(2x^2 + 4x - 7 = 0\).
Transforming your quadratic equation to the standard form ensures that you're ready to apply the quadratic formula, makes the equation clearer, and allows for easier manipulation of terms. In every quadratic problem, looking for and establishing the standard form is always a beneficial first step.

Roots of Quadratic Equation

Roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots are the solutions you find by applying the quadratic formula.
Let's think back to the equation \(2x^2 + 4x - 7 = 0\). Using the quadratic formula, we found two potential roots:

• \(x = \frac{-4 + \sqrt{72}}{4}\)
• \(x = \frac{-4 - \sqrt{72}}{4}\)

Roots can be real or complex numbers depending on the value of the discriminant as discussed earlier.
In essence, to solve a quadratic equation means to find its roots. These roots tell us where the graph of the equation crosses the x-axis. Finding roots is a fundamental skill in algebra, offering insights into the behavior of quadratic relationships.