Problem 40 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. If the solution involves radicals, round to the nearest hundredth. $$4 x^{2}-13 x+3=0$$

Step-by-Step Solution

Answer

The solution to the equation \(4x^2-13x+3=0\) are \(x_1=3\) and \(x_2=0.25\)

Step 1: Identify the coefficients

From the given quadratic equation \( 4x^2-13x+3=0 \), the coefficients are \(a=4\), \(b=-13\), and \(c=3\)

Step 2: Calculate the discriminant

The discriminant is calculated as \( b^2-4ac = (-13)^2 - 4*4*3 = 169 - 48 = 121 \)

Step 3: Apply the quadratic formula

The quadratic formula is \( x =\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), Substituting the values of a, b, and c into the equation, we get: \( x =\frac{-(-13) \pm \sqrt{121}}{2*4} \). That simplifies to: \( x =\frac{13 \pm 11}{8} \)

Step 4: Find the roots

We can now calculate the two solutions for x: \( x_{1}=\frac{13+11}{8}=3 \) and \( x_{2}=\frac{13-11}{8}=0.25 \)

Key Concepts

Solving Quadratic Equations

When we come across a quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are numerical coefficients, we can approach its solution in multiple ways. The most fundamental method is using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This potent formula provides the roots of any quadratic equation, assuming \( a \), \( b \), and \( c \) are real numbers, and \( a \eq 0 \). To employ it, simply plug in the values of your coefficients and follow the arithmetic to obtain \( x \).

In our example, \( 4x^2-13x+3=0 \), the equation has the coefficients \( a=4 \), \( b=-13 \), and \( c=3 \). The application of the quadratic formula leads us to two potential solutions for \( x \), which are the roots of our equation. It's a reliable method that works in every scenario where the equation has a maximum of two real solutions.

Discriminant of a Quadratic Equation

The discriminant is a key player in the quadratic equation game. It is the part of the quadratic formula under the square root, defined as \( b^2-4ac \). The discriminant tells us about the nature of the roots without actually solving the equation. If it's positive, we have two distinct real roots. If it's zero, there's exactly one real root. But if it's negative, brace yourself for complex roots, which won't be real numbers.

For our textbook problem, we computed the discriminant to be \( 121 \), which is positive. Hence, we are assured it has two real solutions. Understanding the discriminant offers valuable insight even before we delve into finding the solutions.

Factoring Quadratics

Factoring is another technique to solve quadratic equations, suited especially when the equation is factorable to the form \( (x-p)(x-q)=0 \) with \( p \) and \( q \) being the roots. In these cases, each factor represents a potential solution of \( x \). Factoring is typically quicker and more straightforward than using the quadratic formula, but it's not always possible. Not all quadratics are easily factorable, especially when dealing with messy or large coefficients.

In the case where quadratic equations can be factored, they reveal the roots directly through the factors. For instance, if we could factor our example equation \( 4x^2-13x+3 \), it would simplify the solving process, bypassing the need for the quadratic formula. Nevertheless, it's essential to recognize when factoring is viable, and when it's simpler to revert to the trusty quadratic formula.

Radicals in Algebra

Radicals often appear in the solution of quadratic equations, typically in the form of square roots when using the quadratic formula. They represent the square root of a number and are denoted by the symbol \( \sqrt{} \). When the square root of a number is not a neat whole number, we encounter an irrational number, which can be an intimidating outcome.

In our task, the discriminant's square root was a perfect square, making things straightforward. But when radicals are not perfect squares, like \( \sqrt{2} \), they symbolize an endless non-repeating decimal that we usually approximate to ease calculations. Understanding how to work with radicals, including their simplification and rationalization, is a fundamental skill in algebra. It helps students in dealing with the precise and approximate solutions of equations where radicals are involved.