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. Challenge Problems. $$(2 x-1)^{2}+6=6(2 x-1)$$

Short answer

The roots of the quadratic equation are \(x = 3.5\) and \(x = 0.5\).

Step-by-step solution

- Expand the Square

First, expand the squared term \(2x-1)^2\) using the identity \(a-b)^2 = a^2 - 2ab + b^2\). The expansion gives \(4x^2 - 4x + 1\).

- Distribute the 6

Distribute the 6 on the right side of the equation to the expression \(2x - 1\), which gives \(12x - 6\).

- Set the Quadratic Equation to Zero

Combine all terms on one side to set the equation to zero: \(4x^2 - 4x + 1 + 6 - 12x + 6 = 0\), resulting in \(4x^2 - 16x + 7 = 0\).

- Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots where \(a = 4\), \(b = -16\), and \(c = 7\).

- Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac = (-16)^2 - 4\cdot4\cdot7 = 256 - 112 = 144\).

- Apply the Discriminant in the Quadratic Formula

Substituting the values into the quadratic formula gives \(x = \frac{-(-16) \pm \sqrt{144}}{2\cdot4}\).

- Simplify to Find the Roots

Simplifying the expression yields the two roots \(x = \frac{16 \pm 12}{8}\), which are \(x = \frac{28}{8}\) and \(x = \frac{4}{8}\).

- Calculate the Exact Roots

This results in the roots \(x = 3.5\) and \(x = 0.5\). Keep in mind the instruction to keep three significant digits.

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are in the form of \( ax^2 + bx + c = 0 \). To use this method, you need to identify the coefficients \(a\text{,} b\text{, and }c\) from the equation. The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here's why it's so useful: When you plug in the values of \(a\text{, }b\text{, and }c\) into the formula, it allows you to calculate the roots of the quadratic equation directly.

For instance, in the given exercise, with \(a = 4\text{, }b = -16\text{, and }c = 7\), after plugging these into the formula, the potential roots are found by performing the arithmetic, ultimately giving the precise answers. This method guarantees a solution when the equation is quadratic, regardless of whether the roots are real or complex numbers.

Factoring Quadratics

Factoring is another technique for solving quadratic equations. This method works best when the quadratic can be broken down into two binomial expressions. Essentially, you are looking for two numbers that multiply to give \(ac\) and add to give \(b\) in the equation \(ax^2 + bx + c\).

However, factoring can be challenging, especially when the equation is not easily factorable. When it's hard to find factors that work, you might use the quadratic formula instead. Nonetheless, if the equation can be factored, it provides a quick and elegant solution. The roots from factoring are the solutions to each of the binomial expressions set to zero and solved for \(x\).

Discriminant of a Quadratic Equation

The discriminant \( \Delta \), given by the formula \(b^2 - 4ac\), is part of the quadratic formula hidden under the square root. It tells us about the nature of the roots without solving the equation. When \(\Delta > 0\), there are two distinct real roots. If \(\Delta = 0\), there is one real root (also called a repeated root). Lastly, if \(\Delta < 0\), the roots are complex and come in conjugate pairs.

In our exercise, the discriminant is 144, which is positive, indicating that two real and distinct roots exist. It is an essential step before proceeding with the quadratic formula, as it can save time by revealing the nature of the roots right away.

Expanding Binomials

Expanding binomials involves applying the distributive property to multiply two binomial expressions. A common example is squaring a binomial, such as \( (a + b)^2 \), which expands to \( a^2 + 2ab + b^2 \). Understanding how to expand binomials is crucial for solving quadratics, as it allows you to convert an equation with a squared term into standard form \(ax^2 + bx + c\), where factoring or applying the quadratic formula then becomes possible.

For clarification, the given exercise starts with \( (2x - 1)^2 \) which is an application of this concept, resulting in \( 4x^2 - 4x + 1 \). Recognizing these patterns of expansion can make solving quadratic equations more straightforward and intuitive.