Math Problem Statement

Solve the equation: 2x^2-4x-7=0 .  Fully simplify all answers, including non-real solutions.

Solution

To solve the quadratic equation 2x24x7=02x^2 - 4x - 7 = 0, we will use the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Step 1: Identify coefficients

From the equation 2x24x7=02x^2 - 4x - 7 = 0:

  • a=2a = 2
  • b=4b = -4
  • c=7c = -7

Step 2: Plug into the quadratic formula

x=(4)±(4)24(2)(7)2(2)x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-7)}}{2(2)}

Simplify each term: x=4±164(2)(7)4x = \frac{4 \pm \sqrt{16 - 4(2)(-7)}}{4} x=4±16+564x = \frac{4 \pm \sqrt{16 + 56}}{4} x=4±724x = \frac{4 \pm \sqrt{72}}{4}

Step 3: Simplify the square root

72=362=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} Thus: x=4±624x = \frac{4 \pm 6\sqrt{2}}{4}

Step 4: Split into two terms

x=44±624x = \frac{4}{4} \pm \frac{6\sqrt{2}}{4} x=1±322x = 1 \pm \frac{3\sqrt{2}}{2}

Final Answer:

x=1+322orx=1322x = 1 + \frac{3\sqrt{2}}{2} \quad \text{or} \quad x = 1 - \frac{3\sqrt{2}}{2}

These are the two simplified solutions. If non-real solutions existed, they would have appeared during the square root calculation. In this case, the solutions are real.


Would you like me to graph the equation or explain the solution in more detail? 😊

5 Related Questions:

  1. How do you derive the quadratic formula from a general quadratic equation?
  2. What is the geometric interpretation of the solutions to 2x24x7=02x^2 - 4x - 7 = 0?
  3. Can quadratic equations have only one solution? If so, under what conditions?
  4. What happens when the discriminant (b24acb^2 - 4ac) is negative?
  5. How can the quadratic formula be used in real-world problems?

Tip:

Always simplify the square root in a quadratic solution as much as possible to ensure your answer is in its most concise form!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Simplifying Square Roots

Formulas

Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Theorems

Properties of square roots
Discriminant analysis in quadratic equations

Suitable Grade Level

Grades 9-10