Problem 18 - 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. $$ x^{2}=9 $$

Step-by-Step Solution

Answer

The possible solutions are 3 and -3.

Step 1: Identify the Equation

We can see that the equation given is \(x^{2} = 9\). We need to solve for 'x'.

Step 2: Taking the square root

We need to take the square root of both sides. Remember there are two solutions, positive and negative. Therefore, the equation \(x^{2}=9\) becomes \(x = \pm \sqrt{9}\).

Step 3: Simplify the expression

Simplify the expression \(\pm \sqrt{9}\) to find the two solutions. Therefore, the equation becomes \(x = \pm 3\), hence x could be 3 or -3.

Key Concepts

Square Root Method

The square root method is a way to solve quadratic equations in the form of \( x^{2} = c \), where \( c \) is a constant. This method is used when the equation does not have a linear term (\(bx\)) or a constant term (\(c\)) other than the one being squared. To use this method, you take the square root of both sides of the equation. Since squaring a number yields a positive result regardless of the original number being positive or negative, taking the square root of both sides results in two possible solutions: one positive and one negative.

It's crucial to include both potential solutions, typically expressed as \( x = \pm \sqrt{c} \). This accounts for the fact that both \( x \) and \( -x \) will square to \( c \). The square root method provides a quick and straightforward approach to finding solutions to certain types of quadratic equations.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and others. The square root of a number \( c \) is written as \( \sqrt{c} \) and represents the value that, when squared, gives back \( c \). When solving quadratic equations like \( x^{2} = c \), if \( c \) is not a perfect square, the solutions will involve a radical expression.

For example, if \( c \) was 8, the solutions would be expressed as \( x = \pm \sqrt{8} \). This could also be simplified by factoring out perfect squares from the radicand—the number inside the radical—to make the expression cleaner, such as \( \sqrt{8} \) simplifying to \( 2\sqrt{2} \). Understanding how to simplify radical expressions is helpful when expressing quadratic solutions that are not perfect squares.

No Real Solution

There are instances when a quadratic equation might have no real solution. This occurs when the value under the square root sign is negative. In mathematics, the square root of a negative number is not a real number; instead, it's considered an imaginary number. The no real solution scenario is important in the context of quadratic equations, as it indicates that the parabola formed by the equation does not intersect the x-axis at any point.

For example, in the equation \( x^{2} = -9 \), attempting to take the square root of -9 will not yield a real number. This is often conveyed in solutions as 'no real solution' or by indicating the result involves an imaginary number \( i \), where \( i = \sqrt{-1} \). Teachers and educational materials emphasize the importance of recognizing when equations will have real solutions versus when they won't, as this is foundational knowledge in algebra.

Quadratic Solutions as Integers

Sometimes, quadratic equations result in solutions that are whole numbers or integers. This typically happens when the constant term \( c \) is a perfect square. A perfect square is a number that can be expressed as the square of an integer.

For instance, the equation \( x^{2} = 9 \) has solutions that are integers because 9 is a perfect square (\( 3^{2} = 9 \)). Therefore, when we apply the square root method, we get \( x = \pm 3 \), which are integer values. Meanwhile, if the constant term \( c \) is not a perfect square, the solutions will be radical expressions that cannot be simplified to integers. Grasping this aspect is beneficial for students as it helps distinguish between when solutions are expected to be integers and when they are presented in radical form.