Math Problem Statement

solve for x: x^2-7x-8=0

Solution

We are given the quadratic equation:

x27x8=0x^2 - 7x - 8 = 0

To solve for xx, we can use the quadratic formula:

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

For the equation x27x8=0x^2 - 7x - 8 = 0, the coefficients are:

  • a=1a = 1
  • b=7b = -7
  • c=8c = -8

Let's substitute these values into the quadratic formula:

x=(7)±(7)24(1)(8)2(1)x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)}

Simplifying the terms inside the square root:

x=7±49+322x = \frac{7 \pm \sqrt{49 + 32}}{2} x=7±812x = \frac{7 \pm \sqrt{81}}{2} x=7±92x = \frac{7 \pm 9}{2}

Now, calculate the two possible values for xx:

  1. x1=7+92=162=8x_1 = \frac{7 + 9}{2} = \frac{16}{2} = 8
  2. x2=792=22=1x_2 = \frac{7 - 9}{2} = \frac{-2}{2} = -1

Thus, the solutions are:

x=8orx=1x = 8 \quad \text{or} \quad x = -1

Do you want more details, or have any questions?

Here are 5 related questions:

  1. What is the quadratic formula, and when is it used?
  2. How do you determine the number of solutions to a quadratic equation?
  3. Can quadratic equations have complex solutions?
  4. How would you solve x27x8=0x^2 - 7x - 8 = 0 by factoring?
  5. What are other methods for solving quadratic equations?

Tip: Always check the discriminant b24acb^2 - 4ac to quickly determine the nature of the solutions (real or complex).

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations

Formulas

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

Theorems

Quadratic Formula Theorem

Suitable Grade Level

Grades 8-10