Math Problem Statement

ײ-7×+10=0 "solve this using quadratic formula"

Solution

To solve the quadratic equation x27x+10=0x^2 - 7x + 10 = 0 using the quadratic formula, we follow these steps.

The quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Where ax2+bx+c=0ax^2 + bx + c = 0 is the standard form of a quadratic equation.

For the given equation x27x+10=0x^2 - 7x + 10 = 0:

  • a=1a = 1
  • b=7b = -7
  • c=10c = 10

Step-by-step solution:

  1. Calculate the discriminant: Δ=b24ac\Delta = b^2 - 4ac Δ=(7)24(1)(10)=4940=9\Delta = (-7)^2 - 4(1)(10) = 49 - 40 = 9

  2. Apply the quadratic formula: x=(7)±92(1)x = \frac{-(-7) \pm \sqrt{9}}{2(1)} x=7±32x = \frac{7 \pm 3}{2}

  3. Solve for the two possible values of xx: x1=7+32=102=5x_1 = \frac{7 + 3}{2} = \frac{10}{2} = 5 x2=732=42=2x_2 = \frac{7 - 3}{2} = \frac{4}{2} = 2

Final Answer:

The two solutions are x1=5x_1 = 5 and x2=2x_2 = 2.

Would you like more details on any step or further explanation?

Here are 5 related questions:

  1. How do you derive the quadratic formula?
  2. What happens if the discriminant is negative?
  3. Can you solve a quadratic equation by factoring?
  4. What is the significance of the discriminant in quadratic equations?
  5. How would the quadratic formula change if a1a \neq 1?

Tip: The discriminant (Δ=b24ac\Delta = b^2 - 4ac) determines the nature of the roots:

  • Δ>0\Delta > 0: Two real and distinct roots.
  • Δ=0\Delta = 0: One real and repeated root.
  • Δ<0\Delta < 0: Two complex roots.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations

Formulas

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

Theorems

Quadratic formula
Discriminant

Suitable Grade Level

Grades 9-10