Math Problem Statement

HOMEWORK quadraticformula worksheet (1).pdf

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Solution

The document you've uploaded contains quadratic equation problems. Here are some solutions based on the problems presented:


Question 1:

Solve 3x23x5=03x^2 - 3x - 5 = 0.

Solution:

We use the quadratic formula x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=3a = 3, b=3b = -3, and c=5c = -5.

  1. Discriminant: Δ=b24ac=(3)24(3)(5)=9+60=69\Delta = b^2 - 4ac = (-3)^2 - 4(3)(-5) = 9 + 60 = 69.
  2. Roots:
    x=(3)±692(3)=3±696x = \frac{-(-3) \pm \sqrt{69}}{2(3)} = \frac{3 \pm \sqrt{69}}{6} So, the exact roots are x=3+696x = \frac{3 + \sqrt{69}}{6} and x=3696x = \frac{3 - \sqrt{69}}{6}.

Question 2:

Solve 3x214x+4=03x^2 - 14x + 4 = 0, giving answers accurate to 2 decimal places.

Solution:

Using the quadratic formula with a=3a = 3, b=14b = -14, and c=4c = 4:

  1. Discriminant: Δ=(14)24(3)(4)=19648=148\Delta = (-14)^2 - 4(3)(4) = 196 - 48 = 148.
  2. Roots:
    x=(14)±1482(3)=14±12.16556x = \frac{-(-14) \pm \sqrt{148}}{2(3)} = \frac{14 \pm 12.1655}{6} Approximate values:
    x114+12.165564.36,x21412.165560.31x_1 \approx \frac{14 + 12.1655}{6} \approx 4.36, \quad x_2 \approx \frac{14 - 12.1655}{6} \approx 0.31

Would you like more detailed explanations for any of these, or should I solve additional problems from the document?

Here are some related questions to consider:

  1. How do you apply the quadratic formula to solve any equation?
  2. What is the significance of the discriminant in quadratic equations?
  3. What methods exist to solve quadratic equations besides the quadratic formula?
  4. How can we factor quadratic equations when the discriminant is a perfect square?
  5. Why are some quadratic solutions left in exact form and others in decimal form?

Tip: Always check the discriminant first, as it tells you whether you'll have real, repeated, or complex roots.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Algebra

Formulas

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

Theorems

Quadratic Formula Theorem

Suitable Grade Level

Grades 9-12