Problem 29 - Algebra 1

Question

Find the value of \(b^{2}-4 a c\) for the equation. $$5 x^{2}+5 x+\frac{1}{5}=0$$

Step-by-Step Solution

Answer

The value of \(b^{2} - 4ac\) or the discriminant for the given quadratic equation is 21

Step 1: Identifying Coefficients

First, identify the values of \(a\), \(b\), and \(c\) in the given equation \(5x^{2} + 5x + 1/5 = 0\). Therefore, \(a = 5\), \(b = 5\), and \(c = 1/5\)

Step 2: Calculating the Discriminant

Calculate the discriminant using the formula \(D = b^{2} - 4ac\) by substituting the identified values of \(a\), \(b\), and \(c\). Thus, \(D = (5)^{2} - 4 * 5 * (1/5)\)

Step 3: Solving for the Discriminant

Solve for \(D\). Thus, \(D = 25 - 4 = 21\)

Key Concepts

Quadratic Formula

The quadratic formula is a key player in the realm of algebra when it comes to finding solutions to quadratic equations. At its core, the formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) represent the algebraic coefficients of the equation in the form \(ax^2 + bx + c = 0\). It practically gives us the solutions for \(x\) by setting the equation to zero and determining where the parabola, which a quadratic equation represents, intercepts the x-axis.
The beauty of the quadratic formula is that it is universal for all quadratic equations. This means that no matter how complex the coefficients might seem, you can plug them into the formula and calculate the roots of the equation. It's like a mathematical Swiss Army knife - a dependable tool that any algebra enthusiast should have at their disposal.

Algebraic Coefficients

Algebraic coefficients are the numerical buddies accompanying the variable terms in an algebraic expression. In the quadratic equation \(ax^2 + bx + c = 0\), the numbers \(a\), \(b\), and \(c\) are the coefficients that give you valuable information about the parabola the equation represents.

• The first coefficient, \(a\), is linked to the parabola's width and direction. If \(a > 0\), the parabola opens upwards, like a happy smile. If \(a < 0\), it flips to frown at us from above.
• The second coefficient, \(b\), has a role in determining the axis of symmetry and the parabola's horizontal position on a graph.
• The third coefficient, \(c\), is where the parabola crosses the y-axis, anchoring it vertically on the graph.

By playing with these coefficients, you essentially sculpt the graph into numerous forms, which can be captivatingly explored through algebra and graphing.

Solving Quadratic Equations

Unraveling the mysteries of a quadratic equation can be a fulfilling adventure. Solving these equations requires you to find the value of \(x\) that satisfies the equation \(ax^2 + bx + c = 0\). There are a few methods at your disposal:

Factoring

This involves breaking down the equation into two binomials. Unfortunately, this method can get especially tricky if the equation doesn't factor nicely.

Completing the Square

Though a bit more involved, completing the square can help you reveal the values of \(x\). It transforms the equation into a perfect square, clearing the path to solutions.

Using the Quadratic Formula

The most reliable technique is the quadratic formula. It saves the day even when factoring seems like decoding hieroglyphs. Despite its daunting appearance, it systematically churns out the roots of the quadratic equation, revealing where the function intercepts the x-axis.

Discriminant Calculation

The discriminant is all about predicting the narrative of quadratic equations before even solving them. Calculated from the algebraic coefficients using the formula \(D = b^2 - 4ac\), it can tell us the nature of the roots the equation holds:

• If \(D > 0\), you'll encounter two distinct real roots, meaning the parabola touches the x-axis at two separate points.
• If \(D = 0\), the equation boasts a single real root. This is where the parabola gently kisses the x-axis just once - at its vertex.
• If \(D < 0\), the equation will have complex roots. In this case, the parabola doesn't intersect the x-axis at all; it either floats above or sinks below, avoiding any contact.

Using our example from the original exercise, \(D = 21\), this positive discriminant indicates that our quadratic equation will have two real and distinct solutions. This makes the discriminant a crucial step in the grand scheme of quadratic equations; it gives us a heads-up on what type of solutions to expect even before we delve into the nitty-gritty of solving the equation.