Math Problem Statement

How do you find the root of -x²-4x+5

Solution

To find the roots of the quadratic equation x24x+5=0-x^2 - 4x + 5 = 0, you can either factor the equation or use the quadratic formula. Since this equation does not factor easily, let's use the quadratic formula:

The quadratic formula is: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where aa, bb, and cc are the coefficients of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. For the given equation x24x+5=0-x^2 - 4x + 5 = 0, we have:

  • a=1a = -1 (coefficient of x2x^2),
  • b=4b = -4 (coefficient of xx),
  • c=5c = 5 (constant term).

Step-by-step solution:

  1. Substitute a=1a = -1, b=4b = -4, and c=5c = 5 into the quadratic formula:

    x=(4)±(4)24(1)(5)2(1)x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-1)(5)}}{2(-1)}

  2. Simplify the expression:

    x=4±16+202x = \frac{4 \pm \sqrt{16 + 20}}{-2}

    x=4±362x = \frac{4 \pm \sqrt{36}}{-2}

  3. Solve the square root of 36:

    x=4±62x = \frac{4 \pm 6}{-2}

  4. Find the two possible solutions:

    • First solution: x=4+62=102=5x = \frac{4 + 6}{-2} = \frac{10}{-2} = -5

    • Second solution: x=462=22=1x = \frac{4 - 6}{-2} = \frac{-2}{-2} = 1

The roots are:

x=5orx=1x = -5 \quad \text{or} \quad x = 1

Let me know if you'd like more details or have any questions!

Here are 5 related questions to expand your understanding:

  1. How would the quadratic formula change if bb were positive?
  2. How do you determine the nature of the roots using the discriminant?
  3. Can you solve this quadratic equation by completing the square?
  4. What does the graph of x24x+5-x^2 - 4x + 5 look like?
  5. How would you solve x24x+5>0-x^2 - 4x + 5 > 0 (an inequality)?

Tip: Always check the discriminant b24acb^2 - 4ac before solving a quadratic equation. It tells you how many and what kind of roots you will get!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Roots of Equations

Formulas

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

Theorems

Quadratic Formula
Discriminant Analysis

Suitable Grade Level

Grades 8-10