Problem 37 - 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. $$ a^{2}+3=12 $$

Step-by-Step Solution

Answer

The solutions to the equation \( a^{2}+3=12 \) are \( a = 3 \) and \( a = -3 \).

Step 1: Isolate \( a^{2} \) on one side of the equation

The aim is to isolate \( a^2 \) on one side of the equation. To do so, subtract 3 from both sides of the equation to get \( a^{2}=12-3 \)

Step 2: Simplify the right side of the equation

Simplify the right side of the equation to get \( a^{2}=9 \).

Step 3: Solve for \( a \)

To find the value of \( a \), take the square root of both sides of the equation. It should be noted that when taking square root of a square, both positive and negative square roots should be considered as solutions to obtain \( a = \pm 3 \).

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra that allows us to find the unknown variable in equations of the form \( ax^2 + bx + c = 0 \). In this specific problem, we're dealing with a simpler version where only the \( a^2 \) term exists alongside constants. The process of solving these equations involves several key steps:

• Isolate the quadratic term: Start by rearranging the equation to get the quadratic term \( a^2 \) on one side. In our case, subtraction was required to isolate \( a^2 \).
• Simplify the equation: Once isolated, focus on simplifying any constants present in the equation. This helps in accurately solving for the variable.
• Find the solutions: By solving for \( a \), you take the square root of both sides. Always remember both positive and negative roots exist since squaring either would return the same original number.

This method ensures you correctly determine the possible solutions for \( a \), which in our example are \( a = \pm 3 \). Understanding this pathway is crucial for tackling more complex quadratic equations in the future.

Radical Expressions

Radical expressions often appear when solving quadratic equations, particularly when exact integer results aren't available. A radical expression includes a root symbol \( \sqrt{} \) and denotes values that are derived from square roots, cube roots, etc.To handle a radical expression correctly:

• Identify perfect squares: If your solution involves a square root, check to see if the number under the root is a perfect square. Perfect squares will simplify to whole numbers, making the equation easier to solve.
• Consider multiple roots: Like in the quadratic equation we solved, radical expressions sometimes yield two solutions: a positive and a negative root. This dual nature comes from the properties of real numbers.
• Express solutions clearly: If numbers cannot be simplified perfectly, express this using radical signs. This allows for precise answers without decimals.

Radical expressions help express exact roots which are valuable in many branches of mathematics, including geometry and calculus.

Real Solutions

Real solutions are solutions to equations that can be plotted on the number line. They are what we often deal with in basic algebra and can either be rational or irrational numbers.When solving quadratic equations, it's important to note:

• Check for a real number: Ensure that the value is not imaginary when taking roots of negative numbers in real contexts.
• Both positive and negative values: In quadratic equations, both values are real solutions in standard cases if they result in a valid number when squared.
• No real solution case: Sometimes, equations will lead to \( \sqrt{-1} \) or similar, indicating no real numbers satisfy the equation. This outcome means the parabola does not intersect the x-axis.

In our exercise, \( a = \pm 3 \) were the real solutions, meaning all values fit nicely on a number line and are perfectly usable in further applications of math you might encounter.