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Chapter 2: Problem 13

Which of the following quadratic equations has no solution? (A) \(0=-2(x-5)^{2}+3\) (B) \(0=-2(x-5)(x+3)\) (C) \(0=2(x-5)^{2}+3\) (D) \(0=2(x+5)(x+3)\)

Short Answer

Expert verified
(C) has no real solution.

Step by step solution

01

- Understand the problem

Identify which quadratic equation has no real solution by analyzing each equation.
02

- Analyze Equation (A)

Equation (A) is: \(0=-2(x-5)^{2}+3\)Rearrange to standard form: \(-2(x-5)^2 = -3\)Divide both sides by -2:\((x-5)^{2} = \frac{3}{2}\)Since \((x-5)^{2}\) can always be made positive, this equation has two real solutions.
03

- Analyze Equation (B)

Equation (B) is: \(0=-2(x-5)(x+3)\)Since this is a product of two different factors (\(x-5\) and \(x+3\)) multiplied by a negative coefficient, it can be zero for certain values of x:\(x = 5\) or \(x = -3\)Thus, this equation has two real solutions.
04

- Analyze Equation (C)

Equation (C) is: \(0=2(x-5)^{2}+3\)Rearrange to standard form: \(2(x-5)^{2} = -3\)Divide both sides by 2:\((x-5)^{2} = -\frac{3}{2}\)Since \((x-5)^{2}\) is always non-negative, it cannot be equal to a negative number. Therefore, this equation has no real solution.
05

- Analyze Equation (D)

Equation (D) is: \(0=2(x+5)(x+3)\)Since this is a product of two different factors (\(x+5\) and \(x+3\)) multiplied by a positive coefficient, it can be zero for certain values of x:\(x = -5\) or \(x = -3\)Thus, this equation has two real solutions.
06

Conclusion

After analyzing each equation, the quadratic equation with no real solution is (C).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving quadratic equations
Quadratic equations are mathematical expressions of the form \[ax^2 + bx + c = 0\], where *a*, *b*, and *c* are coefficients. The solutions to these equations provide the x-values where the equation equals zero.
There are several methods for solving quadratic equations, including:
  • Factoring: Expressing the equation as a product of two binomials.
  • Completing the square: Manipulating the equation into a perfect square trinomial.
  • Quadratic Formula: Using the formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to find the solutions.
Each method converts the quadratic equation into a form that makes it easier to determine the values of *x*. Solving quadratic equations often appears in many standardized tests, such as the SAT. It鈥檚 important to understand each method thoroughly to apply the correct one depending on the problem.
real solutions
Determining if a quadratic equation has real solutions means finding out whether the x-values satisfy the equation. Real solutions can be checked through the discriminant, \(b^2 - 4ac\).
Here鈥檚 how:
  • If the discriminant is positive, there are two distinct real solutions.
  • If the discriminant is zero, there is exactly one real solution (a repeated root).
  • If the discriminant is negative, there are no real solutions (the solutions are complex or imaginary).
For example, in the equation (C) 0 = 2(x-5)虏 + 3, we found no real solution because solving it resulted in a negative number under the square root, indicating no intersection with the x-axis.
SAT math problems
Quadratic equations are a key topic in SAT math problems. They can appear in many forms and require students to apply different methods to solve them. Understanding how to solve these quickly and accurately is crucial for SAT success.
SAT math problems may ask students to:
  • Solve for x in a given quadratic equation.
  • Analyze the properties of the graph of a quadratic function.
  • Determine the number and type of solutions.
Practice with a variety of problems, including those where you must identify if the equation has real solutions, like in exercise (C), is essential. Many resources, including online platforms and textbooks, provide ample practice opportunities.
algebraic analysis
Algebraic analysis refers to breaking down and solving equations methodically. In the context of quadratic equations, it means:
  • Recognizing the form of a quadratic equation.
  • Identifying and carrying out the best method to solve it (factoring, completing the square, or using the quadratic formula).
  • Analyzing the results to determine the type and number of solutions.
Through algebraic analysis, you can determine, as in our example, that equation (C) 0 = 2(x-5)虏 + 3 has no real solution by first rewriting it and then observing that it asks for the square of a number to equal a negative value. Developing good algebraic analysis skills is essential for mastering math at all levels.

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