- A quadratic expression or equation is a polynomial whose highest exponent is 2.
- The root root is the solution to the quadratic equation.
- Using factoring we find the solution of the quadratic equation.
- A quadratic will have a double root if the quadratic is a perfect square trinomial.
Factoring x2 + bx + c
x2 + bx + c = (x + p) (x + q) provided p + q = b and pq =c.
When factoring a trinomial, first consider the signs of p and q.
(x + p)(x + q) | x2+bx+c signs of b and c |
(x+2)(x+3) x2+5x+6 | b is positive; c is positive |
(x+2)(x+(-3)) x2-x-6 | b is negative; c is negative |
(x+(-2))(x+3) x2+5x-6 | b is positive; c is negative |
(x+(-2))(x+(-3)) x2-5x+6 | b is negative; c is positive |
Observing the signs of b and c in the table we see that:
* b and c are positive when both p and q are positive.
* b is negative and c is positive when both p and q are negative.
* c is negative when p and q have different signs.
Examples:
- Factor x2 + 11x + 18
Find two positive factors of 18 whose sum is 11. Make a table
Factors of 18 | Sum of factors | |
18, 1 | 18+1=19 | Not correct |
9,2 | 9 + 2 = 11 | Correct sum |
6, 3 | 6 + 3 = 9 | Not correct |
The factors 9 and 2 have a sum of 11, so they are correct values of p and q.
x2 + 11x + 18 = (x + 9)(x + 2)
- Show that the factors of n2 - 6n + 8 is (n-4)(n-2)
Factors of 8 | Sum of factors | |
-8, -1 | -8+(-1)=-9 | Not correct |
-4,-2 | -4 + (-2) = -6 | Correct sum |
n2 - 6n + 8 = (n-4)(n-2)
- Show that the factors of y2 + 2y - 15 = (y + 5)(y - 3)
Factors of -15 | Sum of factors | |
-15, 1 | -15+1=-14 | Not correct |
15, -1 | 15-1=14 | Not correct |
-5, 3 | -5+3=-2 | Not correct |
5, -3 | 5-3=2 | Correct sum |
y2 + 2y - 15 = (y + 5)(y - 3)
- Solve the equation x2 + 3x = 18
x2 + 3x = 18 ----- original equation
x2 + 3x - 18 = 0 ----- subtract 18 from each side
(x + 6)(x - 3) = 0 ----- factor left side
(x + 6) = 0 or x - 3 = 0
Therefore x = -6 or x = 3 ------ solve for x
The solutions of the equation are -6 and 3
- Solve 6x2 + 42x = 0
6x2 + 42x = 0 ---- original equation
6x(x + 7) = 0 --- factor left side
6x = 0 or x + 7 = 0 ---- solve for x
x = 0 or x = -7
The solutions of the equation are 0 and -7
Directions: Solve the following problems. Also write at least ten examples of your own.
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