Problem 1 - Algebra 2

Question

Find the exact solutions by using the Quadratic Formula. \(8 x^{2}+18 x-5=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{4}\) and \(x = -\frac{5}{2}\).

Step 1: Identify Coefficients

The quadratic equation is given in the standard form \(ax^2 + bx + c = 0\). Identify the coefficients: \(a = 8\), \(b = 18\), \(c = -5\).

Step 2: Write Down the Quadratic Formula

The Quadratic Formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula will be used to find the roots of the given equation.

Step 3: Calculate the Discriminant

The discriminant is \(b^2 - 4ac\). Substitute the values: \(b = 18\), \(a = 8\), and \(c = -5\) to get \(18^2 - 4 \times 8 \times (-5)\). Simplify to find \(324 + 160 = 484\).

Step 4: Calculate the Square Root of the Discriminant

Find the square root of the discriminant \(\sqrt{484}\), which is \(22\).

Step 5: Apply the Quadratic Formula

Substitute \(b = 18\), \(\sqrt{484} = 22\), and \(a = 8\) into the Quadratic Formula: \(x = \frac{-18 \pm 22}{2 \times 8}\).

Step 6: Solve for the Two Possible Roots

Calculate the two possible roots using the \(\pm\) in the formula: \(x_1 = \frac{-18 + 22}{16}\) and \(x_2 = \frac{-18 - 22}{16}\). Simplify to find \(x_1 = \frac{4}{16} = \frac{1}{4}\) and \(x_2 = \frac{-40}{16} = -\frac{5}{2}\).

Key Concepts

Discriminant

In quadratic equations, the discriminant plays a crucial role in determining the nature of the roots. It is found in the expression under the square root of the Quadratic Formula. For the equation \(ax^2 + bx + c = 0\), the discriminant is given by \(b^2 - 4ac\).
This value tells us:

• If the discriminant is positive, the equation has two different real roots.
• If it is zero, there is exactly one real root (also called a repeated or double root).
• If it is negative, the roots are complex or imaginary, meaning there're no real solutions.

Knowing how to calculate the discriminant helps in predicting the type and number of solutions without even solving the equation entirely. For example, in our exercise, the discriminant calculated is 484, which is positive. Thus, we can expect two distinct real roots for the given quadratic equation.

Roots of quadratic equations

Roots of quadratic equations are the solutions that satisfy the equation when substituted for \(x\). In a standard quadratic equation \(ax^2 + bx + c = 0\), roots can be found using different methods such as factoring, completing the square, or using the Quadratic Formula. Each method serves different purposes depending on the equation's coefficients and complexity.

With the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the roots are determined directly from the equation's coefficients without the need to factor or complete the square.
For instance, in our exercise, the roots were calculated by substituting in the coefficients to find \(\frac{1}{4}\) and \(-\frac{5}{2}\), which are the solutions for the equation \(8x^2 + 18x - 5 = 0\). This provides a reliable and straightforward approach to finding the roots regardless of how simple or complex the quadratic equation is.

Solving quadratic equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the given equation. Several methods exist:

• Factoring: Involves expressing the equation as a product of linear factors and then setting each factor to zero. This method works best if the equation is easily factorable.
• Completing the Square: Involves rewriting the equation in the form \((x + p)^2 = q\) and then solving for \(x\). It's useful for deriving the vertex form of a parabola and when the quadratic doesn't factor neatly.
• Quadratic Formula: A universal method given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which can solve any quadratic equation, regardless of factorability.

In our example, using the Quadratic Formula was efficient due to the straightforward calculation of the discriminant (484) and subsequent determination of the roots. It's a method that simplifies the process, making it accessible to solve quadratic equations with any real number coefficients.