Question

Solve the equation using any method. Explain your reasoning. \(x^2+3 x+\frac{5}{4}=0\)

Short answer

The solutions of the equation \(x^2+3 x+\frac{5}{4}=0\) are \(x = -\frac{1}{2}\) and \(x = -\frac{5}{2}\)

Step-by-step solution

Identifying the coefficients

In the given equation \(x^2+3x+\frac{5}{4}=0\), \(a = 1\), \(b = 3\), and \(c = \frac{5}{4}\).

Calculating the Discriminant

Next, calculate the discriminant by using the formula \(b^2 - 4ac\). So here, the discriminant is \(3^2 - 4*1*\frac{5}{4} = 9 - 5 = 4\). A positive number 4 is shown, so it means that there are two distinct real solutions.

Calculating the roots

Now use the quadratic formula \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) which simplifies to \(\frac{-3 \pm \sqrt{4}}{2}\) to find the roots of the equation. The solutions are \(x = \frac{-3 + 2}{2} = -\frac{1}{2}\) and \(x = \frac{-3 -2}{2} = -\frac{5}{2}\).

Key concepts

Discriminant

The discriminant is an essential part of solving quadratic equations. It helps to determine the nature of the roots of the equation. In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by the formula \( \Delta = b^2 - 4ac \). The value of the discriminant provides insights into the number and type of solutions you can expect.

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, which is also known as a repeated or double root.
• If \( \Delta < 0 \), there are no real solutions. Instead, the solutions are complex or imaginary numbers.

In the exercise, the discriminant is calculated as \( 4 \), indicating that there are two distinct real solutions. This insight is valuable because it immediately tells us what kind of solutions we will find even before applying the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). While there are many methods to solve them, using the quadratic formula is universal. The formula is:\[\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's a breakdown of its components:

• \(-b\) denotes the negative of the coefficient \(b\).
• \(\pm \) indicates that there will be two solutions.
• The expression under the square root sign \(\sqrt{b^2 - 4ac}\) is the discriminant.
• Finally, the whole expression is divided by \(2a\), which is twice the coefficient of \(x^2\).

Applying this formula to the exercise, we plug in the values (\(a = 1\), \(b = 3\), \(c = \frac{5}{4}\)) to find the solutions. The resulting computation yields two roots: \(x = -\frac{1}{2}\) and \(x = -\frac{5}{2}\). This method works consistently, regardless of whether the roots are real or complex.

Real Solutions

Real solutions in the context of quadratic equations are solutions that are not imaginary. They can be represented on a number line or as exact points on a graph. The nature of these solutions is primarily determined by the discriminant:

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• These are the values of \(x\) where the parabola (graph of the quadratic equation) intersects the x-axis.

In practical terms, real solutions mean that the equation has actual answers that you can plot or verify. In our exercise, after determining that the discriminant is \(4\) (a positive number), we can conclude that the solutions are real and distinct. This is reflected in the found solutions of \(x = -\frac{1}{2}\) and \(x = -\frac{5}{2}\), both of which are clear real numbers that provide meaningful points of intersection on the graph of the equation \(x^2+3x+\frac{5}{4}=0\). Being aware of whether a quadratic equation's solutions are real helps us understand the behavior and characteristics of the function we are examining.