Problem 16 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$4 x^{2}-12 x+9=0$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{2}\).

Step 1: Identify coefficients

In a quadratic equation of the form \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). For the equation \(4x^2 - 12x + 9 = 0\), we have \(a = 4\), \(b = -12\), and \(c = 9\).

Step 2: Write down the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula is used to find the solutions of any quadratic equation.

Step 3: Substitute coefficients into the formula

Substitute \(a = 4\), \(b = -12\), and \(c = 9\) into the quadratic formula: \(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}\).

Step 4: Simplify inside the square root

Calculate the discriminant: \((-12)^2 - 4 \cdot 4 \cdot 9 = 144 - 144 = 0\). Now the equation becomes: \(x = \frac{12 \pm \sqrt{0}}{8}\).

Step 5: Solve the equation

Since the discriminant is 0, there is only one solution: \(x = \frac{12}{8} = \frac{3}{2}\). Therefore, the solution to the equation \(4x^2 - 12x + 9 = 0\) is \(x = \frac{3}{2}\).

Key Concepts

quadratic formula

The quadratic formula is a powerful tool that allows us to find the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
To use this formula, you merely need to identify the coefficients \(a\), \(b\), and \(c\).
For example, in the equation \(4x^2 - 12x + 9 = 0\), the coefficients are:

• \(a = 4\)
• \(b = -12\)
• \(c = 9\)

By substituting these values into the quadratic formula, you can solve for \(x\). This formula incorporates all the necessary calculations in one expression, making the process systematic and straightforward.

discriminant

The discriminant is a crucial part of the quadratic formula that helps us determine the nature of the roots of a quadratic equation. It is represented by:
\[\text{Discriminant} = b^2 - 4ac\]
The discriminant tells us how many solutions (or roots) the quadratic equation has and what type they are:

• If the discriminant is positive (\( > 0 \)), there are two distinct real roots.
• If the discriminant is zero (\( = 0 \)), there is exactly one real root.
• If the discriminant is negative (\( < 0 \)), there are two complex roots.

In our example, \(4x^2 - 12x + 9 = 0\), the discriminant is:
\[(-12)^2 - 4 \times 4 \times 9 = 144 - 144 = 0\]
Since the discriminant is zero, the equation has one real root.

coefficients

Coefficients are the numerical constants that multiply the variables in an equation. In a quadratic equation of the form \(ax^2 + bx + c = 0\):

• \(a\) is the coefficient of \(x^2\), which we call the quadratic coefficient.
• \(b\) is the coefficient of \(x\), known as the linear coefficient.
• \(c\) is the constant term, also known as the constant coefficient.

Accurately identifying these coefficients is the first step in solving quadratic equations using the quadratic formula.
For the equation \(4x^2 - 12x + 9 = 0\), we have:

• Quadratic coefficient \(a = 4\)
• Linear coefficient \(b = -12\)
• Constant coefficient \(c = 9\)

By substituting these into the quadratic formula, we can find the solution efficiently.