factor the following 2 quadratic equations: 1) x 2 + 6x – 16 = 0 2) 2x 2 + 4x – 10 = 0 the...
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Factor the following 2 quadratic equations:1) x2 + 6x – 16 = 02) 2x2 + 4x – 10 = 0 The second one is more difficult, right? That’s because the roots are not integers We need to solve the relation using the
Quadratic Formula
6.4 - The Quadratic Formula
2x2 + 4x – 10 = 0 2(x2 + 2x – 5) = 0 If we use (x + 1)2, we get x2 + 2x + 1 We need x2 + 2x – 5, so 6 less than above 2((x + 1)2 – 6) = 0 2(x + 1)2 – 12 = 0 2(x + 1)2 = 12 (x + 1)2 = 6 x + 1 = x = -1 Therefore the roots are x=-1+ & x =-1-
You can to solve using Vertex Form too:
Based on form ax2 + bx + c = 0 2x2 + 4x – 10 = 0 a = 2; b = 4; c = -10
Solve Using Quadratic Formula
Quadratic FormulaJust substitute the values for a, b and c into the formula, and
find the 2 values of x
Therefore, the roots of 2x2 + 4x – 10 = 0 are x = and x = x =
Solve Using Quadratic Formula
2x2 + 4x – 10 = 0a = 2; b = 4; c = -10
A rectangular field is going to be completely enclosed by 100m of fencing. Create a quadratic relation that shows how the area of the field will depend on its width. Then, determine the dimensions of the field that will results in an area of 575 m2. Round your answers to 2 decimal places.
Example #2
Let w be the width of the field If total perimeter is 100m, and
width is w, then length must be
Area = length x width = lw A = w(50 – w) A = 50w – w2
575 = 50w – w2
0 = -w2 + 50w – 575 Now use quadratic formula to find roots.
Example #2 cont’d
50 - w
w
0 = -w2 + 50w – 575 a = -1; b = 50; c = -575
Since 50 – w = 50 – 17.9 = 32.1, we can see that w = 17.9m and l = 32.1 m
Example #2 cont’d
The roots of a quadratic equation of the form ax2 + bx + c = 0 can be determined using the quadratic formula:
In Summary…
Quadratic relations can have two, one or no x-intercepts.
6.5 Interpreting Quadratic Equation Roots
This graph shows the quadratic equation:-x2 + x + 6 = 0, and has 2 solutions: x = -
2 and x = 3
One/No Solutions/Root
This graph shows the quadratic equation:x2 – 6x + 9 = 0, and has 1 solution: x = 3
This graph shows the quadratic equation:
2x2 – 4x + 5 = 0, and has no solutions.
Challenge Question
How can you determine the number of solutions to a quadratic equation
without solving it?
The discriminant allows us to determine the number of real roots of each equation
◦ If D > 0, there are 2 real solutions◦ If D = 0, there is 1 real solution◦ If D < 0, there are no real solutions.
EX. 3x2 + 4x + 5 = 0
D < 0, so no solutions.
Discriminant
Determine the number of zeros for
Let’s try to solve by completing the square:
, but we need +17.5 +1.5 =
How can you tell the number to solutions by looking at the equation?
Example #2
Another way to solve: use Quadratic Formula.
(a = -2; b = 16; c = -35)
We cannot square root a negative number, so x does not exist. No solutions.
Example #2 cont’d