Question

the product of two numbers and the difference them are 289 and 20 respectively. find the numbers

Answer 1

Answer:

x = 10 + √389, y = -10 + √389 // x = 29.7231, y = 9.72308

or

x = 10 - √389, y = -10 - √389 // x = -9.72308, y = -29.7231

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Equality Properties

Algebra I

• Solving systems of equations using substitution/elimination
• Standard Form: ax² + bx + c = 0
• Quadratic Formula: $x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}$

Step-by-step explanation:

Step 1: Define Systems

xy = 289

x - y = 20

Step 2: Rewrite Systems

x - y = 20

1. Subtract x on both sides:                   -y = 20 - x
2. Divide -1 on both sides:                      y = x - 20

Step 3: Redefine Systems

xy = 289

y = x - 20

Step 4: Solve for x

Substitution

1. Substitute in y:                    x(x - 20) = 289
2. Distribute x:                         x² - 20x = 289
3. Rewrite [SF]:                        x² - 20x - 289 = 0
4. Define variables:                 a = 1, b = -20, c = -289
5. Substitute [QF]:                   $x=\frac{20\pm\sqrt{(-20)^2-4(1)(-289)} }{2(1)}$
6. Exponents:                          $x=\frac{20\pm\sqrt{400-4(1)(-289)} }{2(1)}$
7. Multiply:                               $x=\frac{20\pm\sqrt{400+1156} }{2}$
8. Add:                                     $x=\frac{20\pm\sqrt{1556} }{2}$
9. Simplify:                               $x=\frac{20\pm 2\sqrt{389} }{2}$
10. Factor GCF:                         $x=\frac{2(10\pm \sqrt{389} )}{2}$
11. Divide:                                 $x=10\pm \sqrt{389}$

Step 5: Solve for y

Possibility 1: x = 10 + √389

1. Define equation:                    x - y = 20
2. Substitute in x:                       (10 + √389) - y = 20
3. Isolate y term:                        -y = 20 - (10 + √389)
4. Isolate y:                                 y = (10 + √389) - 20
5. Combine like terms:              y = -10 + √389
6. Evaluate:                                y = 9.72308

Possibility 2: x = 10 - √389

1. Define equation:                    x - y = 20
2. Substitute in x:                       (10 - √389) - y = 20
3. Isolate y term:                        -y = 20 - (10 - √389)
4. Isolate y:                                 y = (10 - √389) - 20
5. Combine like terms:              y = -10 - √389
6. Evaluate:                                y = -29.7231

Step 6: Identify Solutions

Possibility 1:

x = 10 + √389, y = -10 + √389

x = 29.7231, y = 9.72308

Possibility 2:

x = 10 - √389, y = -10 - √389

x = -9.72308, y = -29.7231