Solve each equation. See Examples 4–6. √4x-x+3=0

Verified step by step guidance

1. First, isolate the square root term on one side of the equation: $\sqrt{4x}=x-3$.
2. Next, square both sides of the equation to eliminate the square root: $(\sqrt{4x}{)}^2=(x-3{)}^2$.
3. Simplify both sides: $4x=x^2-6x+9$.
4. Rearrange the equation to form a quadratic equation: $x^2-10x+9=0$.
5. Solve the quadratic equation using the quadratic formula, factoring, or completing the square to find the values of $x$.

Key Concepts

Square Root Properties

Understanding square root properties is essential for solving equations involving square roots. The square root of a number represents a value that, when multiplied by itself, gives the original number. In this context, isolating the square root term before squaring both sides of the equation is crucial to eliminate the square root and simplify the equation.

Quadratic Equations

The equation presented can be transformed into a quadratic equation, which is typically in the form ax² + bx + c = 0. Quadratic equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Recognizing the structure of a quadratic equation is vital for finding its solutions.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, at least one of the factors must be zero. This principle is often used after factoring a quadratic equation to find its roots. Applying this property allows us to set each factor equal to zero and solve for the variable, leading to the solutions of the original equation.