Problem 65 - Algebra 1: Concepts and Skills

Question

Choose a method and solve the quadratic equation. Explain your choice. $$ 3 x^{2}+5 x-12=0 $$

Step-by-Step Solution

Answer

The solutions of the quadratic equation \(3x^{2}+5x-12=0\) are \(x=-1\) and \(x=4\).

Step 1: Identification of Parameters

Firstly, identify the parameters in the quadratic equation. In this case, \(a=3\), \(b=5\), and \(c=-12\) from the equation \(3x^{2}+5x-12=0\)

Step 2: Calculation of Discriminant

Then, calculate the discriminant which is \(b^{2}-4ac\). Substitute \(a=3\), \(b=5\), and \(c=-12\) into this formula. The result is \(D= (5)^{2} - 4*3*(-12) = 25 +144 = 169\)

Step 3: Application of Quadratic Formula

Now, apply the quadratic formula \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\). Substitute \(a=3\), \(b=5\), and \(D=169\) into this equation, and solve for \(x\). The solutions of this equation will be \(x1=\frac{-5+ \sqrt{169}}{2*3} = -1\) and \(x2=\frac{-5- \sqrt{169}}{2*3}= 4\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a method for finding the roots of the equation, which are the values of \(x\) that satisfy the equation. The formula itself is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Using this formula allows you to find the roots even when factoring the quadratic equation is not straightforward. By directly substituting the coefficients \(a\), \(b\), and \(c\) from the equation into the formula, you can systematically arrive at the root values.
Some key features of the quadratic formula include:

• It works for every quadratic equation, regardless of whether the roots are real, complex, or irrational.
• The \(\pm\) symbol indicates that there are usually two solutions, arising from the two possible values for the square root.

The quadratic formula is thorough and reliable, making it an essential part of your mathematical toolkit.

Discriminant

The discriminant is a crucial component of the quadratic formula. It is the part under the square root: \(b^2 - 4ac\). The discriminant determines the nature and number of the roots of a quadratic equation.
Here's what the discriminant tells us about the roots of the equation:

• If the discriminant is positive, \(b^2 - 4ac > 0\), there are two distinct real roots. This happens because the square root of a positive number gives a positive and a negative result, contributing to two solutions.
• If the discriminant is zero, \(b^2 - 4ac = 0\), there is exactly one real root. Also known as a repeated or double root, it represents the point where the graph of the equation just touches the x-axis.
• If the discriminant is negative, \(b^2 - 4ac < 0\), there are no real roots. Instead, the roots are complex numbers, which means the graph does not intersect the x-axis.

Understanding the discriminant helps you predict the behavior of the roots before doing any calculations.

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions you find by setting the quadratic expression equal to zero and solving for \(x\). They represent the x-values where the graph of the quadratic equation touches or crosses the x-axis.
The quadratic equation \(ax^2 + bx + c = 0\) might have:

• Two distinct real roots, if the discriminant \(b^2 - 4ac\) is positive.
• One real root if the discriminant equals zero, meaning the graph is tangent to the x-axis.
• Two complex roots if the discriminant is negative, indicating no real solutions.

In the original example, the quadratic equation \(3x^2 + 5x - 12 = 0\) had a discriminant of 169, which is positive. This resulted in two distinct real roots, \(x = -1\) and \(x = 4\). These roots illustrate where the function crosses the x-axis in a graph.
Understanding roots is pivotal as they provide critical points that describe the solution space of quadratic equations.