Problem 35 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible. $$3 x^{2}-12 x+12=0$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 2 \). As the discriminant was zero, it confirms the solutions are real and equal.

Step 1: Identify coefficients

First, identify the values of \( a \), \( b \), and \( c \) from the given equation. So here are the variables: - \( a = 3 \)- \( b = -12 \)- \( c = 12 \)

Step 2: Calculate the discriminant (\( \Delta \))

The discriminant is used to find out the nature of the roots of the quadratic equation. Compute the value of the discriminator using the formula \( \Delta = b^{2} - 4ac \), which ends up being \( \Delta = (-12)^{2} - 4*3*12 = 144 - 144 = 0 \). The result is zero, hence, the nature of the roots will be real and equal.

Step 3: Calculate the roots of the equation

Calculate the roots using the general quadratic formula: \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \). With \( a = 3 \), \( b = -12 \), and (\( \Delta = 0 \)), we apply these values into the formula and get \( x = \frac{-(-12) \pm \sqrt{0}}{2*3} \), which simplifies to \( x = 2 \).

Key Concepts

Discriminant in Quadratic Equations

The discriminant of a quadratic equation provides valuable insights into the nature and type of the equation's roots. When presented with an equation of the form \( ax^2 + bx + c = 0 \), the discriminant is calculated using the formula \( \Delta = b^2 - 4ac \).

The discriminant can yield three types of results:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, or more technically, two real and equal roots.
• If \( \Delta < 0 \), the equation has two complex roots.

In our exercise, the quadratic equation is \( 3x^2 - 12x + 12 = 0 \). Using the given coefficients, \( a = 3 \), \( b = -12 \), and \( c = 12 \), we calculate the discriminant as \( \Delta = (-12)^2 - 4(3)(12) \), which simplifies to \( \Delta = 0 \). This indicates that the equation has real and equal roots, an important clue for predicting the solution's nature before actually solving the equation.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations and is expressed as \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients of the equation \( ax^2 + bx + c = 0 \), and \( \Delta \) is the discriminant.

This formula is versatile because it directly provides the equation's roots regardless of the discriminant's value, although knowing the discriminant's value will foretell whether the roots are real and distinct, real and equal, or complex. For the given problem, we've already computed \( \Delta = 0 \), hence when we substitute the values into the quadratic formula, the term \( \sqrt{\Delta} \) becomes \( \sqrt{0} \), which is 0. Consequently, the equation simplifies, making the calculation of the roots straightforward. In this case, the solution to the equation becomes \( x = \frac{12}{6} \), which means that our single, real, and equal root is \( x = 2 \).

Real and Equal Roots

When a quadratic equation has a discriminant value of zero, it implies that the equation has real and equal roots. This condition signifies that the parabola represented by the quadratic equation touches the x-axis at exactly one point.

In our example, the discriminant \( \Delta = 0 \) tells us that the roots are not only real but are also equal—there is only one unique solution to the equation. It is as if the quadratic equation folds onto itself at the vertex, resulting in a single x-intercept on the graph. After applying the values to the quadratic formula, we establish that the root for the equation \( 3x^2 - 12x + 12 = 0 \) is \( x = 2 \). This singular solution is also the vertex of the parabola and dubbing this solution as 'real and equal' emphasizes the fact that both roots (\( x_1 \) and \( x_2 \)) coincide.