1. Factor it.
12x^2 - 10x - 42 = 0
12x^2 - 28x + 18x - 42 = 0
4x(3x - 7) + 6(3x - 7) = 0
(4x + 6)(3x - 7) = 0
x = {-6/4, 7/3} = {-3/2, 7/3}
2. Use the quadratic equation.
X = (-b ±√(b^2-4ac))/(2a)
= (-(-10) ±√((-10)^2 - 4(12)(-42)))/(2(12)) (substitute a=12, b=-10, c=-42)
= (10 ±√(100 + 2016))/24 (evaluate)
= (5 ± 23)/12 (evaluate, reduce)
x = {-3/2, 7/3} (evaluate, reduce)
3. Complete the square. (Equivalent to using the quadratic equation.)
12x^2 - 10x - 42 = 0
x^2 - 5/6x = 7/2 (add 42, divide by 12, reduce)
x^2 - 5/6x + 25/144 = 7/2 + 25/144 (add the square of half of 5/6)
(x - 5/12)^2 = 529/144 (show the completed square, evaluate right side)
x - 5/12 = ±23/12 (square root)
x = (5 ± 23)/12 (add 5/12, same answer as above)
Other techniques can also be used to solve a quadratic, including
- graph it
- use an iteration technique, such as Newton's iteration - use trial and error, with the Rational Root Theorem roots as starting points.
12x^2 - 10x - 42 = 0
12x^2 - 28x + 18x - 42 = 0
4x(3x - 7) + 6(3x - 7) = 0
(4x + 6)(3x - 7) = 0
x = {-6/4, 7/3} = {-3/2, 7/3}
2. Use the quadratic equation.
X = (-b ±√(b^2-4ac))/(2a)
= (-(-10) ±√((-10)^2 - 4(12)(-42)))/(2(12)) (substitute a=12, b=-10, c=-42)
= (10 ±√(100 + 2016))/24 (evaluate)
= (5 ± 23)/12 (evaluate, reduce)
x = {-3/2, 7/3} (evaluate, reduce)
3. Complete the square. (Equivalent to using the quadratic equation.)
12x^2 - 10x - 42 = 0
x^2 - 5/6x = 7/2 (add 42, divide by 12, reduce)
x^2 - 5/6x + 25/144 = 7/2 + 25/144 (add the square of half of 5/6)
(x - 5/12)^2 = 529/144 (show the completed square, evaluate right side)
x - 5/12 = ±23/12 (square root)
x = (5 ± 23)/12 (add 5/12, same answer as above)
Other techniques can also be used to solve a quadratic, including
- graph it
- use an iteration technique, such as Newton's iteration - use trial and error, with the Rational Root Theorem roots as starting points.