Problem 43 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ 5 x^{2}+5=20 $$

Step-by-Step Solution

Answer

The solutions to the given equation are \(sqrt(3)\) and \(-sqrt(3)\)

Step 1: Rewrite the Equation in Standard Form

Rearrange the terms to obtain the standard form of the equation \(ax^{2} + bx + c = 0\). Here the equation becomes \(5x^{2} - 20 + 5 = 0\). By simplifying it, we get: \(5x^{2} - 15 = 0\)

Step 2: Factor Out Any Common Factors

Before moving on with the quadratic formula, find out if there's a common factor that could be factored out. Here, 5 is a common factor. So, removing 5, the equation becomes \(x^{2} - 3 = 0\)

Step 3: Apply the Quadratic Formula

The quadratic formula is \(x = [-b ± sqrt(b² - 4ac)]/2a\). Here 'a' is 1 (after removing the common factor), 'b' is 0 and 'c' is -3. So, the quadratic formula gives two values for x: \(x₁ = sqrt(3)\) and \(x₂ = -sqrt(3)\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are mathematical expressions of the second degree. Typically presented as \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), it provides a systematic approach to finding the roots of any quadratic equation, real or complex.
After evaluating the coefficients of the terms in the standard form, you substitute them into the quadratic formula to solve for \(x\). In our exercise example, the equation \(5x^2 + 5 = 20\) can be simplified to \(x^2 - 3 = 0\) before applying the formula. By placing \(a = 1\), \(b = 0\), and \(c = -3\) we can solve for the roots, yielding \(x_1 = \sqrt{3}\) and \(x_2 = -\sqrt{3}\). These solutions are the points where the graph of the equation intersects the x-axis.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential starting point for solving these equations, and it is expressed as \(ax^2 + bx + c = 0\), with \(a\), \(b\), and \(c\) representing constants.
In the context of our exercise, the standard form is reached by rearranging the terms of the equation \(5x^2 + 5 = 20\) to achieve \(5x^2 - 15 = 0\). This form highlights the relationship between the coefficients \(a\), \(b\), and \(c\), and is the basis from which we can factor or apply the quadratic formula for solving the equation. Whenever you're given a quadratic equation, ensure that you rearrange it into this standardized format to streamline the solving process.

Factoring in Algebra

Factoring is an algebraic method used to simplify expressions and solve equations. In quadratics, if the equation can be written as a product of two binomials, factoring is often the fastest way to find the roots.
For instance, in our problem, factoring out the common factor of 5 leads to the simpler equation \(x^2 - 3 = 0\). From this form, you can observe the structure of the quadratic and potentially factor it further into binomials to find the roots. Although in this case, the equation cannot be factored easily using integers, factoring is still a critical step that reduces the equation to its simplest form before applying the quadratic formula.

Radical Expressions

Radical expressions are mathematical expressions that involve roots, such as square roots, cube roots, and so forth. They appear frequently when solving quadratic equations, especially when the solutions include irrational numbers.
The solutions to the exercise \(x^2 - 3 = 0\) are \(x = \sqrt{3}\) and \(x = -\sqrt{3}\), both of which are radical expressions. When a quadratic equation does not have integer solutions, writing them as radical expressions provides an exact representation of the roots. Understanding how to simplify and manipulate radical expressions is essential in algebra, as it allows you to work with a precise form of irrational solutions without resorting to estimates or decimal approximations.