Problem 18 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$12 x^{2}+23 x-9=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{3}\) and \(x = -\frac{9}{4}\).

Step 1 - Identify the Quadratic Equation Standard Form

The equation given is \(12x^2 + 23x - 9 = 0\), which is in the standard form of a quadratic equation: \(ax^2 + bx + c = 0\), where \(a = 12\), \(b = 23\), and \(c = -9\).

Step 2 - Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Substitute \(a = 12\), \(b = 23\), and \(c = -9\) into the formula:\[\Delta = 23^2 - 4 \times 12 \times (-9) = 529 + 432 = 961\]

Step 3 - Check the Nature of the Roots

The discriminant \(\Delta = 961\) is greater than zero and is a perfect square (since \(31^2 = 961\)). Thus, the quadratic equation has two distinct real roots.

Step 4 - Use the Quadratic Formula

The roots of the quadratic equation \(ax^2 + bx + c = 0\) are given by:\[x = \frac{{-b \pm \sqrt{\Delta}}}{2a}\]Substitute \(b = 23\), \(\Delta = 961\), and \(a = 12\):\[x = \frac{{-23 \pm 31}}{24}\]

Step 5 - Calculate the Two Roots

Calculate each root:\[x_1 = \frac{{-23 + 31}}{24} = \frac{8}{24} = \frac{1}{3}\]\[x_2 = \frac{{-23 - 31}}{24} = \frac{-54}{24} = -\frac{9}{4}\]

Step 6 - Conclusion

The solutions to the quadratic equation \(12x^2 + 23x - 9 = 0\) are \(x = \frac{1}{3}\) and \(x = -\frac{9}{4}\).

Key Concepts

Quadratic Formula

When we talk about solving quadratic equations, the quadratic formula is one of the most powerful and general methods to use. It provides a universal solution to any quadratic equation by calculating its roots. The formula is expressed as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]Here, the symbols \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation's standard form: \(ax^2 + bx + c = 0\). To utilize the formula efficiently:

• Identify the coefficients \(a\), \(b\), and \(c\).
• Calculate the discriminant, \(b^2 - 4ac\).
• Plug these values into the quadratic formula to find the roots.

This method works for any quadratic equation, whether it can be factored easily or not. With this formula, you can solve any quadratic equation without first needing to factor or complete the square.

Discriminant

In the context of quadratic equations, the discriminant plays a crucial role in understanding the nature of the roots without even solving the equation fully. The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\). Analyzing the value of the discriminant allows us to determine the type of solutions we can expect.

• If \(\Delta > 0\) and is a perfect square, the equation has two distinct real roots.
• If \(\Delta > 0\) but not a perfect square, the roots are real and irrational.
• If \(\Delta = 0\), the equation has exactly one real root (also known as a double root).
• If \(\Delta < 0\), the roots are not real but complex or imaginary.

Understanding the discriminant thus gives us an early insight into whether we'll find real or complex roots without actually calculating them.

Real Roots

When solving quadratic equations, discovering that it has real roots can simplify further calculations considerably. Real roots arise when the discriminant (\(b^2 - 4ac\)) turns out to be zero or positive. Depending on the discriminant value:

• Two distinct real roots occur if \(\Delta > 0\).
• A double real root appears if \(\Delta = 0\).

If the quadratic equation opens up or down symmetrically about the vertex, these roots are the points where the curve intersects with the x-axis. Real roots are often more straightforward to handle since they don't involve complex numbers, which can prove more challenging to interpret practically.

Quadratic Equation Standard Form

The standard form of a quadratic equation is specific and structured, which makes it easier to apply different solving techniques, including using the quadratic formula, factoring, or completing the square. The standard form of a quadratic equation is:\[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are real number coefficients, and \(a eq 0\) (since otherwise, the equation wouldn't be quadratic but linear). Identifying and writing a quadratic equation in its standard form allows students to:

• Directly apply the quadratic formula.
• Recognize patterns for factoring.
• Calculate its discriminant easily.

This format's consistency is key for methodical problem solving, helping learners grasp quadratic equations more intuitively.