Textbook Question
Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 = 2x - 5
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1. Equations & Inequalities
The Quadratic Formula
2:36 minutesProblem 61
Textbook Question
Solve each equation in Exercises 60–63 by the square root property. x^2/2 + 5 = -3
Verified step by step guidance1
Isolate the term containing the variable squared. In this case, subtract 5 from both sides of the equation to get: \(\frac{x^2}{2} = -8\).
Multiply both sides of the equation by 2 to eliminate the fraction: \(x^2 = -16\).
Apply the square root property, which states that if \(x^2 = a\), then \(x = \pm\sqrt{a}\). Here, you would take the square root of both sides: \(x = \pm\sqrt{-16}\).
Recognize that the square root of a negative number involves an imaginary number. Rewrite \(\sqrt{-16}\) as \(\sqrt{16} \cdot \sqrt{-1}\), which simplifies to \(4i\), where \(i\) is the imaginary unit.
Conclude that the solutions to the equation are \(x = 4i\) and \(x = -4i\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Root Property
The square root property states that if x^2 = k, then x = ±√k. This property is essential for solving quadratic equations, as it allows us to isolate the variable by taking the square root of both sides. It is particularly useful when the equation is in the form of x^2 = a constant, enabling straightforward solutions.
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Isolating the Variable
Isolating the variable involves rearranging an equation to get the variable on one side and all other terms on the opposite side. This step is crucial in solving equations, as it simplifies the problem and allows for the application of methods like the square root property. In the given equation, isolating x^2 is the first step before applying the square root property.
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Quadratic Equations
Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They can have zero, one, or two real solutions, depending on the discriminant (b^2 - 4ac). Understanding the nature of quadratic equations is vital for applying various solving techniques, including factoring, completing the square, and using the square root property.
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