Problem 57 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. If it is true, give a reason. If it is false, give a counterexample. \(x^{2}=c\) has two solutions when \(c>0\)

Step-by-Step Solution

Answer

The statement is true. When \(c>0\), the quadratic equation \(x^{2}=c\) indeed has two distinct solutions.

Step 1: Understand the Quadratic Equation

Any quadratic equation \(x^{2}=c\) is easily rearranged to the form \(x^{2}-c=0\). This is a simple quadratic equation, where \(a=1, b=0\), and \(c=-c\) in the general quadratic formula \(ax^{2} + bx + c=0\). The general formula to find roots of a quadratic equation is \(x=\frac{-b±\sqrt{b^{2}-4ac}}{2a}\). Here, the term under the square root, \(b^{2}-4ac\), is known as the discriminant, and the number of solutions for the equation depends on its value.

Step 2: Analyse the Discriminant and Number of Solutions

The discriminant of the equation is calculated as \(b^{2} - 4ac = 0^{2} - 4*1*(-c) = 4c\). If \(c>0\), then 4c is positive, implying that the discriminant is greater than 0. That means the quadratic equation will have two distinct real solutions.

Step 3: Conclude the Statement

The statement is true because when \(c>0\), \(x^{2}=c\) has two distinct solutions as per the discriminant analysis.

Key Concepts

Discriminant Definition

When working with quadratic equations, an essential concept is the discriminant. It is the part of the quadratic formula under the square root sign, represented by the symbol 'D'. In mathematical terms, for the general quadratic equation \(ax^2 + bx + c = 0\)), the discriminant is calculated as \(D = b^2 - 4ac\)).

The value of the discriminant gives us vital information about the nature and number of solutions to the quadratic equation. If \(D > 0\)), the equation has two distinct real solutions. When \(D = 0\)), there is exactly one real solution, which is also called a repeated or double root. Lastly, if \(D < 0\)), the equation has no real solutions; instead, it has two complex solutions which are conjugates of each other.

Understanding the discriminant is critical because it helps us predict the outcome of solving the quadratic equation without actually finding the exact values of the solutions. It's a quick diagnostic tool to determine the type of roots the equation possesses.

Quadratic Formula

The quadratic formula is a powerful tool that gives the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\)). The formula is given by \(x = \frac{-b \pm \sqrt{D}}{2a}\)), where 'D' is the discriminant \(b^2 - 4ac\)). This formula is derived from completing the square on the general equation and provides the roots directly.

To apply the quadratic formula, you substitute the coefficients \(a\)), \(b\)), and \(c\)) from your equation into the formula. Using the plus sign in \(\pm\)) will give you one root, and using the minus sign will give you the other. Remember that the discriminant \(D\)) inside the square root sign will tell you the nature of these roots - whether they're real or complex.

The quadratic formula is so valuable because it is a standardized method that will always work for any quadratic equation, assuming you follow the correct steps and pay attention to positive and negative signs.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. Given a quadratic equation in standard form \(ax^2 + bx + c = 0\)), there are several methods for finding its solutions: factoring, using the square roots, completing the square, and employing the quadratic formula.

When an equation can be easily factored, this method is often the quickest. However, not all quadratic equations can be factored simply. If the equation is in the form \(x^2=c\)) and \(c\)) is a non-negative number, taking square roots after isolating \(x^2\)) on one side gives two solutions, \(x = \sqrt{c}\)) and \(x = -\sqrt{c}\)).

In cases where the quadratic cannot be factored or is not conducive to the square root method, the quadratic formula remains a fail-safe way to find solutions. It's derived for the general form of the quadratic equation and can handle any scenario, including when the equation has complex solutions. Remember, as demonstrated in the exercise, to always consider the discriminant, as it not only influences the method you might choose but also gives insight into the nature of the equation's roots.