Solve each equation in Exercises 15–34 by the square root property. (8x - 3)² = 5

검증된 단계별 안내

Step 1: Start by isolating the squared term. The equation is already in the form \((8x - 3)^2 = 5\), so no further rearrangement is needed.

Step 2: Apply the square root property. The square root property states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\). Using this property, take the square root of both sides: \(8x - 3 = \pm \sqrt{5}\).

Step 3: Solve for \(8x\) by adding 3 to both sides of the equation: \(8x = 3 \pm \sqrt{5}\).

Step 4: Isolate \(x\) by dividing both sides of the equation by 8: \(x = \frac{3 \pm \sqrt{5}}{8}\).

Step 5: The solution can be expressed as two values: \(x = \frac{3 + \sqrt{5}}{8}\) and \(x = \frac{3 - \sqrt{5}}{8}\). These are the two solutions to the equation.

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주요 개념

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Square Root Property

The square root property states that if a quadratic equation is in the form (x - a)² = b, then the solutions can be found by taking the square root of both sides. This results in two possible equations: x - a = √b and x - a = -√b. This property is essential for solving equations that can be expressed as perfect squares.

Isolating the Variable

Isolating the variable involves rearranging an equation to get the variable on one side and the constants on the other. In the context of the square root property, this means ensuring that the squared term is alone on one side of the equation before applying the square root. This step is crucial for correctly applying the square root property.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using various methods, including factoring, completing the square, and using the quadratic formula. Understanding the structure of quadratic equations is vital for applying the square root property effectively.

Solve each equation in Exercises 15–34 by the square root property. (8x - 3)2 = 5