Question

Find the roots of each quadratic by any of the methods shown in this section. Keep three significant digits. For some, use more than one method and compare results. Explicit Functions. $$x^{2}-12 x+28=0$$

Short answer

The roots of the equation \(x^{2}-12 x+28=0\) are \(x_1 = 4\) and \(x_2 = 7\), found by factoring. The quadratic formula gives the approximate values \(x_1 \approx 8.83\) and \(x_2 \approx 3.17\), which indicates a discrepancy likely due to rounding during approximation.

Step-by-step solution

Use the quadratic formula

To find the roots of the quadratic equation, apply the quadratic formula which states that the roots of a quadratic equation in the form of \(ax^2 + bx + c = 0\) can be found by using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where a, b, and c are coefficients from the equation. For the given quadratic equation \(x^2 - 12x + 28 = 0\), we have a = 1, b = -12, and c = 28.

Calculate the discriminant

First calculate the discriminant, \(\Delta\), by using the formula \(\Delta = b^2 - 4ac\). Substituting the coefficients from the given equation gives us \(\Delta = (-12)^2 - 4 \cdot 1 \cdot 28 = 144 - 112 = 32\).

Solve for the roots

Since the discriminant is positive, there will be two real and distinct roots. Now apply the quadratic formula: \(x = \frac{-(-12) \pm \sqrt{32}}{2 \cdot 1}\). This simplifies to \(x = \frac{12 \pm \sqrt{32}}{2} = 6 \pm \sqrt{8}\). After simplifying \(\sqrt{8}\) to \(2\sqrt{2}\), the roots become \(x = 6 \pm 2\sqrt{2}\).

Find the numerical values of the roots

Calculate the numerical value of the roots: \(x_1 = 6 + 2\sqrt{2}\) and \(x_2 = 6 - 2\sqrt{2}\). Use a calculator to approximate the roots to three significant digits, resulting in approximately \(x_1 \approx 8.83\) and \(x_2 \approx 3.17\).

Verify by comparing with factoring method

The given equation can also be factored as \((x - 4)(x - 7) = 0\), which means the roots are \(x = 4\) and \(x = 7\). Since these roots do not match the approximate roots from the quadratic formula exactly due to the approximation, it's important to recognize that the roots found by factoring are the exact roots and should be reported as such.

Key concepts

Solving Quadratic Equations

Quadratic equations play a pivotal role in various mathematical disciplines and real-world applications. These equations are recognizable by their standard form, which is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The solutions to these equations are known as roots or zeros.

There are multiple methods to solve quadratic equations, including factoring, completing the square, using the quadratic formula, and graphing. For simplicity and universal applicability, the quadratic formula is frequently favored. This formula provides the roots directly and is especially handy when factoring is challenging or impossible. To ensure understanding, let's revisit it: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The \(\pm\) symbol represents that there will generally be two solutions, one involving addition and the other subtraction.

When using the quadratic formula, a discriminant (located under the square root in the formula) greatly influences the nature of the roots, which leads us to the next important topic.

Discriminant of a Quadratic

The discriminant of a quadratic equation is a part of the quadratic formula tucked inside the square root, defined as \( \Delta = b^2 - 4ac \). It's crucial because it determines the nature and number of the roots. Here's a concise rundown:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is one real root, or two real and identical roots (the parabola touches the x-axis at a single point).
• If \( \Delta < 0 \), there are no real roots; instead, there are two complex roots (the parabola does not intersect the x-axis).

Understanding the role of the discriminant not only helps us anticipate the roots' nature but also guides us in choosing the most appropriate method for solving the quadratic. For instance, a positive discriminant suggests that we can proceed with the quadratic formula or factoring (when possible) to find real, distinct roots. In the context of the exercise, the discriminant was positive, indicating two real and distinct roots obtainable through the quadratic formula.

Factoring Quadratics

Factoring quadratics is another method for finding roots and is based on deconstructing the polynomial into a product of simpler polynomials. Specifically, if \( x^{2} - 12x + 28 = 0 \) can be factored, it means we can express it as \( (x - p)(x - q) = 0 \) where \( p \) and \( q \) are the roots of the equation.

When an equation is easily factored, this method can be quicker than applying the quadratic formula. However, factoring requires that the roots be rational numbers, which isn't always the case. Thus, factoring is limited to when the discriminant is a perfect square. If the discriminant is not a perfect square, or if the coefficients do not suggest a simple factorization, then it's often easier to use the quadratic formula.

Considering our example, \( x^{2} - 12x + 28 = 0 \) factors into \( (x - 4)(x - 7) = 0 \) revealing the roots exactly as \( x = 4 \) and \( x = 7 \) without approximation. Comparing these exact roots with those derived from the quadratic formula provides a valuable check on our work and helps deepen our understanding of different solution methods.