Problem 8 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. Write your solutions in simplest form. $$x^{2}-2 x-15=0$$

Step-by-Step Solution

Answer

The solutions for the equation \(x^{2}-2x-15=0\) are \(x = 5\) and \(x = -3\)

Step 1: Identify a, b, and c

Here, the quadratic equation is presented as \(x^{2} - 2x - 15 = 0\). This can be written in the general form \(ax^{2} + bx + c = 0\), where \(a = 1\), \(b = -2\), and \(c = -15\).

Step 2: Apply the Quadratic Formula

Now the quadratic formula \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) will be used to find the values of \(x\). So, substituting \(a = 1\), \(b = -2\) and \(c = -15\) in the formula, we get \(x = \frac{-(-2) \pm \sqrt{(-2)^{2} - 4(1)(-15)}}{2(1)}\)

Step 3: Simplify the expression

Simplify the expression under the square root, then calculate the values in denominator and numerator to get the possible values of \(x\). Hence, \(x = \frac{2 \pm \sqrt{4 + 60}}{2} = \frac{2 \pm \sqrt{64}}{2} = \frac{2 \pm 8}{2}\)

Step 4: Find the solutions

Calculate the values of \(x\). This will give \(x = 5\) or \(x = -3\) as the solution.

Key Concepts

Understanding a Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. It is typically written in the standard form:

• \(ax^2 + bx + c = 0\)

where \(a\), \(b\), and \(c\) are constants and \(a\) is not equal to zero. This equation represents a parabola when graphed on a coordinate plane. The quadratic term \(ax^2\), the linear term \(bx\), and the constant term \(c\) each play a pivotal role in the shape and direction of the parabola.
When solving quadratic equations, our goal is to find the value(s) of \(x\) that satisfy the equation. These values of \(x\) are referred to as the 'roots' or 'solutions' of the quadratic equation. Sometimes, a quadratic equation may have two, one, or no real solutions. The outcome depends on the discriminant \(b^2 - 4ac\) within the quadratic formula.

Solving Quadratic Equations with the Quadratic Formula

The quadratic formula is an essential tool used for solving quadratic equations. It allows you to find the solutions even if factoring is difficult or impossible. The formula is written as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The components of the formula include:

• \(-b\) - This is the opposite of the coefficient of the linear term.
• \(\pm\) - Indicates that we will solve for two possible values of \(x\).
• \(\sqrt{b^2 - 4ac}\) - This part is called the discriminant.
• \(2a\) - This is the denominator common to both possible solutions.

To apply this formula, first identify \(a\), \(b\), and \(c\) from the equation. Substitute these values into the formula to find the solutions. If the discriminant is positive, there are two different real solutions. If it equals zero, there is one real solution. If it's negative, the solutions are complex.

Simplifying Expressions to Find Solutions

Simplifying expressions is a crucial step in using the quadratic formula. Once you plug in \(a\), \(b\), and \(c\) into the formula, you'll encounter an expression under the square root, known as the discriminant \(b^2 - 4ac\). Simplify this expression correctly to ensure you get accurate solutions.
The next step involves calculating the square root of this simplified expression, then substituting the result back into the formula. From there, simplify further to find the potential solutions for \(x\). For example, after computing the discriminant in our problem, \( \sqrt{64}\), you find it's \(8\). Substitute \(8\) back into the expression:

• \(x = \frac{2 \pm 8}{2}\)

Finally, split into two separate calculations:

• \(x = \frac{2 + 8}{2} = 5\)
• \(x = \frac{2 - 8}{2} = -3\)

This simplification process leads us to the two solutions of the quadratic equation. Proper simplifications ensure the solutions are correct and in their simplest form.