The solutions to the equations in order are
1. x = [tex]\frac{-1}{5}[/tex], x= -1,
2. x = [tex]\frac{1}{2}[/tex], x = -3,
3. x = 4 + √10, x= 4 - √10, 4.
4. x = 8 + √40/ 2, x = 8 - √40 / 2 and
6. x= 1/2 + √5/3, x= 1/2 - √5/3.
Step-by-step explanation:
Step 1; For a quadratics equation ax² + bx + c = 0, the product of the solution terms is a×c and the product of the terms is b.
For 5x² + 6x + 1 = 0, the sum is 6 and the product is 5. The terms are 5 and 1.
5x² + 6x + 1 = 5x² + 5x + x + 1 = 5x (x + 1) + 1 (x + 1) = (5x + 1)(x + 1) = 0.
5x + 1 = 0 and x + 1 = 0 so x = [tex]\frac{-1}{5}[/tex], x= -1.
Step 2; For 2x² + 5x - 3 = 0, the sum is 5 and the product is -6. The terms are 6 and -1.
2x² + 5x - 3 = 2x² - 1x + 6x - 3 = x (2x - 1) + 3 (2x - 1) = (x + 3)(2x - 1) = 0.
x + 3 = 0 and 2x - 1 = 0 so x = [tex]\frac{1}{2}[/tex], x = -3.
Step 3; For x² + 2x - 8 = 0, the sum is 2 and the product is -8. The terms are 4 and -2.
x² + 2x - 8 = x² - 2x + 4x - 8 = x (x - 2) + 4 (x - 2) = (x + 4)(x - 2) = 0.
x + 4 = 0 and x - 2 = 0 so x = -4 and x = 2.
Step 4; For equations that cannot be seperated into its factors.Any quadratic equation of the form, ax² + bx + c = 0 can be solved using the formula x = -b ± √b² - 4ac / 2a. Here a, b, and c are the coefficients of the x², x, and the numeric term respectively.
x² - 8x + 6, here a = 1, b = -8, c = 6. x = -b ± √b² - 4ac / 2a = -(-8) ± √(-8)² - 4(1)(6) / 2(2) = 8 ± √64 - 24/2. So x= 8 ± √40 / 2 so x = 8 + √40/ 2, x = 8 - √40 / 2
Step 5; 9x² - 6x - 4, here a = 9, b = -6, c = -4. x =-b ± √b² - 4ac / 2a =-(-6) ± √(-6)² - 4(9)(-4) / 2(9) = 6 ± √36 + 144/18. 180 can also be written as 5 × 36 and √36 = 6. So x= 1/2 + √5/3, x= 1/2 - √5/3.
Answer:
1. x = , x= -1,
2. x = , x = -3,
3. x = 4 + √10, x= 4 - √10, 4.
4. x = 8 + √40/ 2, x = 8 - √40 / 2 and
6. x= 1/2 + √5/3, x= 1/2 - √5/3.
