Example Card: Solving $ax^2 + c = 0$
Solve incomplete quadratics of the form ax^2 + c = 0 by isolating x^2 and taking square roots of both sides. Practice Grade 9 quadratic solving.
Key Concepts
Before you can take the square root, you have to clear away the other numbers. Let's practice isolating the squared variable first.
Example Problem Solve the equation $5x^2 125 = 0$.
Step by Step 1. To solve for $x$, we first need to isolate the $x^2$ term. $$ 5x^2 125 = 0 $$ 2. Use the Addition Property of Equality to move the constant term to the other side. $$ \begin{aligned} 5x^2 125 &= 0 \\ +125 &= +125 \\ \hline 5x^2 &= 125 \end{aligned} $$ 3. Use the Division Property of Equality to isolate $x^2$. $$ \begin{aligned} \frac{5x^2}{5} &= \frac{125}{5} \\ x^2 &= 25 \end{aligned} $$ 4. Now that the equation is in the form $x^2 = a$, take the square root of both sides. $$ \sqrt{x^2} = \pm\sqrt{25} $$ 5. Simplify to find the final solutions. $$ x = \pm 5 $$ 6. Check both solutions : $$ \begin{aligned} 5x^2 125 &= 0 \\ 5(5)^2 125 &\stackrel{?}{=} 0 \\ 5(25) 125 &\stackrel{?}{=} 0 \\ 125 125 &= 0 \quad \checkmark \end{aligned} \quad \begin{aligned} 5x^2 125 &= 0 \\ 5( 5)^2 125 &\stackrel{?}{=} 0 \\ 5(25) 125 &\stackrel{?}{=} 0 \\ 125 125 &= 0 \quad \checkmark \end{aligned} $$.
Common Questions
What are the key steps to solving $ax^2 + c = 0$?
Identify the equation type, isolate the variable using inverse operations, and verify by substituting back into the original equation.
What common mistakes occur when solving $ax^2 + c = 0$?
Applying operations to only one side, sign errors when moving terms, and not checking solutions in the original equation.
How is this skill applied in real problems?
These techniques model physical, financial, and geometric situations where unknown quantities must be found from given conditions.